LIGHTNING SURGE PROPAGATION IN OVERHEAD LINES AND GAS INSULATED BUS-DUCTS AND CABLES by |^LEE KAI-CHUNG B.ScT, Uni v e r s i t y of Wisconsin, 1973 M.Sc., Univ e r s i t y of B r i t i s h Columbia, 1975 M.A.Sc., Univ e r s i t y of B r i t i s h Columbia, 1977 r A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY The Faculty of Graduate Studies i n the Department '_ of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Jul y , 1980 0 Lee Kai-Chung, 1980 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of ____________________ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1WS Date ABSTRACT The propagation c h a r a c t e r i s t i c s of l i g h t n i n g surges i n compressed SFg gas insulated power substation was studied using an electromagnetic transients program. Numerical models were developed to represent the behaviour of d i f f e r e n t system components e s p e c i a l l y under l i g h t n i n g over-voltage conditions. The c h a r a c t e r i s t i c s of l i g h t n i n g surge propagation i n overhead multi-phase untransposed transmission l i n e s was analysed f i r s t . Modal an a l y s i s , :tpgether with s p e c i a l r o t a t i o n techniques to f i t time domain solutions were then used to simulate the wave propagation i n multi-phase untransposed l i n e i n an electromagnetic transients program. Non-linear voltage-dependent corona attenuation and d i s t o r t i o n phenomena were also investigated. Available f i e l d test r e s u l t s could be duplicated to within 5%. The c h a r a c t e r i s t i c s of l i g h t n i n g surge propagation in multi-phase single-core SF^ cables was studied next. A program was developed to obtain the cable parameters for t y p i c a l cable configurations. The amount of core current returning through i t s own sheath and through the earth were computed to i l l u s t r a t e the single phase cable representation for wave propagation i n si n g l e core SFft cables. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i ACKNOWLEDGEMENTS v i INTRODUCTION 1 CHAPTER 1 - LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1. Introduction 3 2. Lightning discharge mechanism 3 3. S t a t i s t i c a l c h a r a c t e r i s t i c s of l i g h t n i n g strokes 7 4. Frequency of l i g h t n i n g strokes to earth 9 5. Frequency of l i g h t n i n g strokes to power l i n e s . . 12 6. Shielding f a i l u r e phenomenon of l i g h t n i n g strokes 13 CHAPTER 2 - LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES 1. Introduction 18 2. Modal analysis for N-phase untransposed l i n e . . . 19 3. Rotation of eigenvectors f o r zero shunt conductance 22 4. Confirmation of accuracy of eigenvalue and eigenvector subroutine 25 5. Real-valued frequency-independent transformation matrix 25 6. Frequency dependent e f f e c t i n l i g h t n i n g surge propagation 27 7. Determination of surge impedance of the struck phase of a transmission l i n e 33 8. Single phase representation for close-by strokes on double c i r c u i t e d l i n e 36 i i i CHAPTER 3 - LIGHTNING WAVE PROPAGATION IN SF & GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM 1. Introduction 46 2. Formation of series impedance matrix f o r SF, cables 49 6 3. Calcu l a t i o n of s e l f and mutual earth return. . . . 50 4. Calcu l a t i o n of s e l f impedance matrix f o r sin g l e core cable. 61 5. Sheath current return c h a r a c t e r i s t i c s f o r usual earth 67 6. Sheath current return c h a r a c t e r i s t i c s f o r substation earth with grounding g r i d network . . . 74 7. Formation of shunt admittance matrix for SFg cables 79 8. Confirmation of numerical accuracy f o r cable parameter c a l c u l a t i o n and current return r a t i o . . 80 9. Single phase representation parameters f o r multi-phase SF^ cables 80 10. Wave propagation i n SF^ cables 84 CHAPTER 4 - CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES 1. Introduction 87 2. Ph y s i c a l properties of corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s 87 3. Transmission l i n e equations for coronated l i n e s . . 90 4. Solution of l i n e equations by compensation method with trapezoidal rules 91 5. Influence on corona by adjacent sub-conductors i n the same bundle 97 6. Influence on corona by adjacent phase conductors . 99 7. Optimal lumping locations and number of corona branch legs 99 8. Overa l l numerical modelling f o r corona e f f e c t s . . 102 i v CHAPTER 5 - CONCLUSIONS 106 APPENDIX A - SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY 107 BIBLIOGRAPHY 109 v ACKNOWLEDGEMENT I would l i k e to show my deepest appreciation to my thesis supervisor Professor Hermann W. Dommel for providing such a unique chance in doing research i n electromagnetic transients of power systems. Dr. Dommel's valuable c r i t i c i s m , and countless hours of discussion during research are also g r a t e f u l l y acknowledged. I am g r a t e f u l to the B.C. Hydro Engineers, Messrs. Jack Sawada, Brent Hughes, Ken Nishikawara and Nick Cuk for t h e i r h e l p f u l discussions. I am also thankful to my fellow graduate students and colleagues Messrs. Obed Abledu, But-Chung Chiu, Shi Wei for t h e i r d i f f e r e n t points of view. The f i n a n c i a l support from the Un i v e r s i t y of B r i t i s h Columbia i n form of teaching and research a s s i s t a n t s h i p and fellowship i s very much appreciated. The f i n a n c i a l assistance of the Systems Engineering D i v i s i o n of the B r i t i s h Columbia Hydro and Power Authority through a Power Systems Research Agreement, and the B r i t i s h Columbia Telephone Company Graduate Scholarship are also g r a t e f u l l y acknowledged. Special thanks are expressed to Miss G a i l Hrehorka i n the E l e c t r i c a l Engineering Main O f f i c e for producing t h i s e x c e l l e n t l y typed t h e s i s . F i n a l l y , I am indebted to my parents for t h e i r continuous encouragement and my wife for her patience. v i 1 INTRODUCTION Every year, atmospheric l i g h t n i n g discharges cause numerous disturbances and damages to e l e c t r i c power systems, such as, destroying transformers and causing black-outs of large areas. This thesis i s devoted to the analysis of l i g h t n i n g surge propagation into compressed SFg gas-insulated substations. The MICA project of the B r i t i s h Columbia Hydro and Power Authority was chosen as a t e s t example. Insulation co-ordination requirements are usually derived from simulated surge propagation studies. This thesis shows that the present p r a c t i c e of i n s u l a t i o n co-ordination design can be improved with the numerical models developed i n t h i s t h e s i s . The contributions of t h i s thesis to i n s u l a t i o n co-ordination design and r e l a t e d power system studies includes the following: 1. Determination of wave propagation i n untransposed l i n e s - Analysis i s used, with a s p e c i a l r o t a t i o n of modal parameters and transformation matrices to make the method sui t a b l e f or time-domain solutions of wave propagation i n multi-phase untransposed l i n e . The s u i t a b i l i t y of d i f f e r e n t s i m p l i f i e d transmission l i n e models i s c l a r i f i e d by comparing simulation r e s u l t s with those from an exact multi-phase rep re sentat ion. 2. Representation of non-linear voltage-dependent corona e f f e c t s - Corona d i s t o r t i o n and attenuation has been simulated with voltage dependent 44 12 v e l o c i t i e s and c o r r e c t i o n factors i n the past ' , or with f i n i t e d ifference methods. However, these methods are i n e f f i c i e n t for d i g i t a l computer a p p l i c a t i o n s . More e f f i c i e n t computational algorithms, using compensation methods, are developed i n t h i s thesis to 2 investigate the non-linear voltage dependent corona e f f e c t s . 3. Determination of wave propagation i n multi-phase si n g l e core SF^ cables - Published methods for the c a l c u l a t i o n of cable constants give inconsistent r e s u l t s . A new cable constants program f o r m u l t i -phase single core SF^-cables has been developed by the author, using various converging.Mnfinite s e r i e s . The complete s h i e l d i n g e f f e c t of the exter n a l l y grounded sheath at frequencies above 1 k Hz has been confirmed with t h i s program. The problem of transient groundrise caused by i n t e r n a l breakdowns or by l i g h t n i n g impulses, as studied by Ontario Hydro"^, i s not included i n t h i s t h e s i s . These transient p o t e n t i a l d i f f e r e n c e s between SF^ bus-ducts and ground occur mainly at the junction with the overhead l i n e . As t h i s thesis shows, the current return i n the SF bus-duct i s completely o through the sheath at frequencies above 1 kHz, whereas the current return of the l i g h t n i n g impulse on the overhead l i n e i s i n the ground. At the junction, the return current must therefore pass from the sheath into the ground through the ground leads, which i n turn causes the transient groundrise problem. These transient groundrises are an important factor i n the design of the grounding system, because they can cause damage to ru a u x i l i a r y wiring or shocks to personnel. 3 CHAPTER 1: LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1. Introduction The f i r s t important experiment on l i g h t n i n g was done by Benjamin F r a n k l i n , who used f l y i n g k i t e s to show that l i g h t n i n g i s e l e c t r i c a l i n nature. For more than two centuries, l i g h t n i n g has been the subject of acti v e research. Much of t h i s research has been concerned with the pro-te c t i o n of people and property against the e f f e c t s of l i g h t n i n g stroke. 2. Lightning discharge mechanisms Lightning strokes are f i r s t i n i t i a t e d i n s i d e thunder-clouds. A thunder-cloud usually contains several negative and p o s i t i v e charge centres d i s t r i b u t e d i n d i f f e r e n t locations as shown i n Figure 1.1a. As soon as the e l e c t r o n i s j u m p i n g o v e r t o n e u t r a l i z e the p o s i t i v e charge, a step leader s t a r t s to move down the earth i n d i s c r e t e zig-zag steps of about 50 meters i n length as shown i n Figure 1.1b. This downward p i l o t stroke i s about 1 cm i n diameter and i s not v i s u a l l y detectable by the human eye. As t h i s stepped leader continues to progress downwards, p o s i t i v e charges are induced and accumulated on the ground surface. Eventually, these p o s i t i v e charges jump upwards and form the return stroke to meet the downward stepped leader as shown i n Figure 1.1c. This highly luminous return stroke produces most of the thunder which i s heard. The return stroke i s about 10 cm i n diameter and at a temperature of around 30,000°K. Once a l l the p o s i t i v e charges transfer to the thunder-cloud as shown i n Figure l . l d , the discharged charge centre completely becomes 4 a. Charge n e u t r a l i z a t i o n b. Stepped leader moving within the cloud. downwards. I n i t i a l i z a t i o n of upward d. Complete upward propagation moving return leader. of return leader to cloud (charge centre becomes p o s i t i v e ) . Figure 1.1: Charge d i s t r i b u t i o n and propagation during i n i t i a l l i g h t n i n g discharge. 5 a. Discharge between 2 b. Negative charge dart charge centres. stroke flowing down the continuous earth path. c. Negative charge dart d. Formation of subsequent stroke about to h i t return leader from ground the ground. to cloud charge centres. Figure 1.2: Charge d i s t r i b u t i o n and propagation during sub-sequent dart leader '(multipli-stroke l i g h t n i n g ) . 6 1 p o s i t i v e and s i n g l e stroke l i g h t n i n g discharge i s completed. 2 3 However, about 50% of a l l l i g h t n i n g flashes are multi-strokes ' and contain 3 or 4 subsequent strokes, t y p i c a l l y separated by 30 to 40 ms. About les s than 100 ms a f t e r the f i r s t stroke, a high p o t e n t i a l d i f f e r e n c e i s again established between the charge centres. Discharges again occur and a dart leader i s formed which moves earthwards i n the previous main stream as shown i n Figure 1.2a to 1.2c. S i m i l a r l y , a return stroke i s also formed and more p o s i t i v e charges trans f e r to the thunder-clouds as shown i n Figure 1.2d. The whole process of multi-strokes with r e l a t i v e stroke magnitudes and time scales i s i l l u s t r a t e d i n Figure 1.3a and- 1.3b. T y p i c a l sub-sequent strokes are of the i n i t i a l stroke magnitude and are w e l l separated (about 30 - 40 ms) i n time. The i n i t i a l stroke i s the prime factor i n the i n s u l a t i o n co-ordination studies, but subsequent strokes must be taken into account as a r r e s t e r s must be able to handle r e p e t i t i v e discharges, and the dead times of the auto-reclosing switchgear must be set longer. V e l o c i t y 100% = 300 m/jjs Figure 1.3a: Diagram showing time i n t e r v a l s between i n i t i a l and subsquent strokes.(Ref.3) 100|is lOQuis -35ms- •35ms Figure 1.3b: Current magnitudes of i n i t i a l and subsequent strokes i n t y p i c a l l i g h t n i n g flashes. 3. S t a t i s t i c a l c h a r a c t e r i s t i c s of l i g h t n i n g strokes Due to the d i f f e r e n t d i s t r i b u t i o n s and i n t e n s i t i e s of charge centres ins i d e the thunder-cloud, the c h a r a c t e r i s t i c s of l i g h t n i n g strokes show a large s t a t i s t i c a l v a r i a t i o n i n both magnitude and shape. a. Magnitude of l i g h t n i n g strokes The voltage stress on the power system depends on the magnitude of the l i g h t n i n g current, which i s therefore a c r i t i c a l 4 52 factor i n determining i n s u l a t i o n requirements. -Recorded measurements .' are shown i n Figure 1.4. I t can be seen that 80% of the l i g h t n i n g current magnitudes are within 10 to 100 kA, and only 5% exceed magnitude of 100 kA. 52 It i s suggested that the l i g h t n i n g stroke has to be simulated as an incident current source to the power l i n e with a maximum current magnitude of 100 kA for i n s u l a t i o n co-ordination studies. However, t h i s current source w i l l become an overvoltage wave when propagating down the power l i n e due to the inherent surge impedance of the l i n e . Thus, for 8 Figure 1.4: Cumulative P r o b a b i l i t y of Occurrence of the Amplitudes of Lightning Currents obtained by summarising r e s u l t s from more than 624 measured incidents from 9 countries (Ref. 4). 9 equipment te s t purposes, overvoltage waves are prescribed, b. Waveshape of l i g h t n i n g stroke The l i g h t n i n g waveshapes measured by d i f f e r e n t researchers essen-t i a l l y resemble a double exponential waveshape of d i f f e r e n t r i s e time and decay time. The observed spread of r i s e time i s from very short to 10 us. 5 The observed decay time also spreads from 2 to 100 ps (see Figure 1.5a). The e l e c t r i c power industry therefore agreed many years ago to use a l i g h t n i n g overvoltage wave for equipment i n s u l a t i o n t e s t i n g purpose of a shape 1.2 x 50 ys (explanation-of designation i n Figure 1.5b and 1.5c). Some te s t i n g p r e c r i p t i o n s also specify that t h i s f u l l wave be chopped with a spark gap i n the t a i l to expose the equipment to the higher frequencies which are contained i n the voltage collapse. 4. Frequency of l i g h t n i n g strokes to earth The thunderstorm a c t i v i t y on earth i s measured by the isokeraunic l e v e l . This isokeraunic l e v e l (IKL) gives the number of days per year that thunder has been heard. Usually, thunder cannot be heard outside a 7-24 km radius. An updated world map of isokeraunic l e v e l i s shown i n Figure 1.6. As expected, higher IKL i s found within the t r o p i c a l and sub-tropical regions close to the equator. A f t e r obtaining the IKL of a given place, the number of strokes 2 7 to earth per km (N) i n a p a r t i c u l a r l o c a t i o n i s given by N = A (IKL) stroke/(km 2 - yr) where A = 0.1 to 0.2 10 t^=rise time =time to crest t2=decay time =time to h a l f value 10 100 T i m e ^ s ) a. Wave fronts and t a i l s of l i g h t n i n g surges Time Double exponential wave V = v^ ( e - * t e r ) s wave Time to crest t 1 =1.67(X2-X ) . =1.2jus Time to h a l f value t =X.-X I 4 o =50JJS Time c. T y p i c a l surge waveform impulse generator Figure 1.5: Waveshape of l i g h t n i n g strokes.(Ref.5) Figure 1.6: World D i s t r i b u t i o n of Thunderstorm Days (Ref.6) 12 5. Frequency of l i g h t n i n g strokes to power l i n e s For estimating the number of l i g h t n i n g strokes to power l i n e s , we can start, from the ' e l e c t r i c a l shadow' cast on the ground by the t a l l tower structure with power l i n e s . The frequency of l i g h t n i n g strokes on the ' e l e c t r i c a l shadow' i s assumed to be the frequency of strokes to the power l i n e s . The width (w) of the shadow area estimated by reference 6 i s chosen. For a power l i n e with two ground wires, the width i s given by (see Figure 1.7) w = 4h + b where h = height of ground wire i n m b = separation between ground wires S i m i l a r l y , f o r a power l i n e with only 1 ground wire, the width i s given by w = 4h where h = height of ground wire i n m and for power l i n e s without ground wires, the width i s given by w = 4h + b where h = height of phase wire i n m b = separation between outermost phase wires Thus, the number of strokes/km - yr to the power l i n e (N ) i s Li N L = 0.1 (IKL) stroke /km -yr (1.1) 13 For a t y p i c a l 500 kV tower of the MICA Dam Project, for the l i n e close to the substation, we have AT fr\ i w o r > \ 4 x 37.5 + 18.64 L = ( O - 1 ^ 3 0 ) loOO ( 1 ' 2 ) = 0.5 strokes /km - yr GW GW GW = ground wire M . O : PW PW = phase wire 2h b 2h $ s = s h i e l d i n g angle = 23° h = height of ground wire = 37.5 m b = width between ground wire = 18.64 m Figure 1.7: Lightning stroke ' E l e c t r i c a l shadows' of a t y p i c a l 500 kV transmission l i n e . 6. Shielding f a i l u r e phenomenon of l i g h t n i n g strokes As shown i n Figure 1.7, ground wires are designed f o r s h i e l d i n g of the phase wire from d i r e c t l i g h t n i n g strokes. However, l i g h t n i n g strokes 14 could s t i l l 'sneak' through the ground wire and h i t the phase wire. Such shielding f a i l u r e s have been recorded i n various countries for d i f f e r e n t tower conf igurat ions. Maikopar^ derived a shielding f a i l u r e curve based on observed f i e l d data (see Figure 1.8). However, the graph does not shows the f a c t that shie l d i n g f a i l u r e s occur mainly on lower l i g h t n i n g s t r i k e currents. At higher currents, (e.g. >14.2;kA f o r MICA), the phase wire i s e f f e c t i v e l y shielded from l i g h t n i n g strokes. As seen from Figure 1.1 and 1.2, the p i l o t downward stepped leader from the thundercloud i s formed and propagates earthward f r e e l y regardless of the structure on earth i n i t i a l l y . Later, the return strokes i s formed from a ground object closest to the leader t i p , (ground wire, phase wire, or the ground) arid propagates upward to meet the stepped leader to complete the l i g h t n i n g path. This ground object i s the object which w i l l be struck by the l i g h t n i n g stroke. 8 Brown analysed r e s u l t s from the 120,000 km - yr l i n e i n the Path-finder Project and deduced that the target i s not chosen u n t i l the distance between the stepped leader t i p and the prospective object i s shorter than the s t r i k i n g distance r . This s t r i k i n g distance i s related only to the stroke current as 7 i T 0 • 7 5 r g = 7.1 I m where I = current in kA From t h i s s t r i k i n g distance concept, we can develop the e l e c t r o -geometric model, as shown i n Figure 1.9a. The shielding f a i l u r e of ground-wire at lower current amplitudes can c o r r e c t l y explained by t h i s more ref i n e d method. The degree of exposure of d i f f e r e n t conductors i s 15 SHIELD ANGLE - DEGREES Figure 1.8: P r o b a b i l i t y of Shielding F a i l u r e vs. Shield Angle between Ground Wire and Top Phase Conductor/ represented by drawing exposure arcs of s t r i k i n g distance radius, and centred at each i n d i v i d u a l conductors. The i n i t i a l power frequency voltage of the phase conductor i s ignored as t h i s voltage i s comparatively small to the discharge voltage of the l i g h t n i n g strokes. For l i g h t n i n g currents of 10 kA and 14.2 kA, the corresponding exposure of the phase conductor PW i s shown. It can be seen that the phase conductor exposure to l i g h t n i n g stroke i s decreased with increases i n s t r i k i n g current. For current of amplitudes higher than 14.2 kA, for t h i s tower structure i n MICA project, the phase conductor i s e f f e c t i v e l y shielded by the ground wire and the ground as the exposure arc i s n e g l i g i b l e i n s i z e (See Figure 1.9a). 16 BC=phase wire exposed arc for s h i e l d i n g f a i l u r e SI Figure 1.9a: Electrogeometric model with maximum s t r i k i n g distance of 53.3m. Stroke Current in kA Figure 1.9b: Frequency D i s t r i b u t i o n of Shielding F a i l u r e Stroke Currents in case of Shielding F a i l u r e 17 Brown et a l further investigated t h i s s i t u a t i o n by taking the angular d i s t r i b u t i o n of the l i g h t n i n g stroke g(^) into account and evaluated the phase wire exposed arc for d i f f e r e n t stroke currents as x = r h s i n i i ^ l ^ 2 2 where: g(^) = - cos y (1.4) our The d e t a i l e d a n a l y t i c a l r e s u l t s are shown in Figure 1.9b. For MICA tower of maximum s t r i k i n g distance of 53.3 m (I =14.2kA), the r e s u l t shows that l e s s than 1% of sh i e l d i n g f a i l u r e l i g h t n i n g currents to the phase wire w i l l exceed 14 kA. This agrees well with the geometric i n t e r p r e t a t i o n shown in Figure 1.9a. The l i g h t n i n g stroke usually h i t the ground wire or the tower. In t h i s case, a voltage w i l l b u i l d up ."across the in s u l a t o r because of the po t e n t i a l r i s e on the tower crossarms. If the insula t o r f l a s h over (' backflashover') to the phase conductors, then l i g h t n i n g overvoltage surges w i l l appear on the conductors. 18 CHAPTER 2 : LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES 1. Introduct ion Propagation of l i g h t n i n g surges due to d i r e c t strokes, or backflash-overs i n overhead l i n e s influences the choice of i n s u l a t i o n r e q uire-ments. One must know the attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s of the l i n e i n order to f i n d the overvoltages entering the substation where most of the equipment i s concentrated. This section t r i e s to answer the questions whether i t i s possible to represent untransposed overhead l i n e s as equivalent single phase l i n e s for the s t r i c k e n conductor with accuracy, and whether s e l f , p o s i t i v e or zero sequence impedances should be used in such single-phase representations? At f i r s t , f i e l d t e s t s r e s u l t s are duplicated by using a Fourier 9 transformation method. This method not only includes the frequency-dependence of the l i n e parameters, but i t also uses the exact complex, frequency-dependent transformation matrix which requires recomputation at each frequency within the frequency range.typical of l i g h t n i n g surges (e.g. 10 k Hz to 1MHz). This method i s recommended for the simulation of distant strokes where the frequency dependent c h a r a c t e r i s t i c s must be included. For close-by l i g h t n i n g strokes, the above frequency-domain solution can be replaced by a simpler time-domain solution method. This method i s based on modal analysis with frequency-independent parameters and r e a l -valued transformation matrices. The r e s u l t s obtained with the simpler time-domain simulation method agree very well ( < 4% deviation) with the accurate frequency-domain simulation method. After confirming the correctness in the time-domain simulation with the exact N - phase representation of the overhead l i n e f o r close-by l i g h t n i n g strokes, the r e s u l t s obtained are thus compared against s i n g l e -phase approximate representations as presently used. Furthermore, a d d i t i o n a l recommendations are made on how to remove unce r t a i n t i e s i n the choice of surge impedance values of overhead l i n e s . ^ I t i s also found that frequency dependence ef f e c t of nearby l i g h t n i n g stroke can be ignored, Line parameters can be chosen at high frequency e.g. at 1 M Hz, and l i n e 11 12 resistance can be ignored as contradictory to the previous f i n d i n g s . ' 2. Modal analysis for N - phase untransposed l i n e The well known transmission l i n e equations describe the propagation of electromagnetic waves on overhead transmission l i n e s . However, contrary to the s i n g l e phase case, the solution to the N- phase case cannot be obtained e a s i l y since each of the N overhead conductors i s mutually coupled to the other conductors. The following two sets of simultaneous second-order p a r t i a l d i f f e r e n t i a l matrix equations describing the change in voltages and currents along the N - phase l i n e must be solved: dVJ phase" dx n x l r phase r phase " 1 Jnxn L J n x l (2.1) d l dx phase" n x l = r Y P h a s e i r v p h a s e i L Jnxn L J n x l (2.2) where [ z p h a s e ] . [Yphase-| [-,-phase j nxn nxn n x l [V phase. J n x l impedance matrix in phase domain admittance matrix i n phase domain phase current vector phase voltage vector 20 The N coupled differential equations in equations (2.1) and (2.2) can.be transformed into N decoupled equations by replacing phase quan-t i t i e s with modal quantities, [ vphas e ] = [ T ] [ vmod e ] (2.3) [ Iphase ] = [ T _ 3 [Imode] (2.4) and by choosing [T ] and [T ±] in a certain way, as described later. Applying equations (2.3) and (2.4) to equations (2.1) and (2.2) gives demode dx [ T , ] "1 [ z P h a S e ] [T . ] [ I m ° d e ] (2.5) _ j-^ mode^ |.^.mode-| (2.6) and d l dx mode = [ I . ] " 1 [ Y P h a S e ] [T ] [ .V n o d e ] (2.7) = j-Ymode^ ^ o d e ^ (2.8) To find x , and replace T^ ], we f i r s t differentiate equation (2.1) with respect to *" V phase] with equation (2.2): d l 1 dx d V h a s e l d x 2 ^phase-j j.Yphase^ ^phase^ (2.9) With equation (2.3), this can be written in modal quantities as 2„mode dx = t y " 1 [ Z p h a s e ] [ Y p h a S e ] [T v] 1^°**] (2.10) (2.11) I f [T 1 i s the matrix of eigenvectors of I Z p h a S e ] [ y p h a s e ] , then [ A] v becomes a diagonal matrix, with i t s elements being the eigenvalues of j-zphase^ ^ p h a s e ^ S i m i l a r l y , f o r the current q u a n t i t i e s , we have J2Tmode a L_ L dx 2 = [ T . ] - 1 [ Y P H A S £ ] [ Z p h a S e ] [T.] [ I m ° d e ] (2.12) - UJ [ I m ° d e ] (2.13) where [T\] = matrix of eigenvectors of [Y*\ ] [7? ], with [A] being i d e n t i c a l to that i n equation (2.11). Taking the transpose of the expression for [A] i n equation (2.12) and comparing i t with that f o r [A] i n equation (2.10), while remembering that [ Z p ^ a s e ] and [ y P ^ a s e ] are symmetric, gives: [A] = [?±f [ Z p h a S e ] [ Y P H A S £ ] ( [ T . ] ^ " 1 = I T v ] _ 1 t Z P h a S £ ] [ Y P H A S E ] [ T V ] or [T v] = (-[T.] 1")" 1 (2.14) Thus, only one of the matrices or [T ] i s needed. Using only , rm -i • i . i J i <r rr,niode, . r,,mode, the [T^J-matrix, we can obtain the modal parameters of [Z J and [Y J from equation (2.6) as [ Z m ° d e ] = [7±f [ Z p h a S e ] [T.] (2.15) and from equation (2. 8) as [ Y M ° D E ] = [ I . ] - 1 [ Y P h a S e ] ( [ T . ] 1 1 ) - 1 (2.15a) r mode,-! _ t r v p h a s e , - l r m , or [ Y ] = [T.] [Y ] [T.] (2.16) 13 14 In the computer pugram developed for t h i s modal a n a l y s i s , ' equation (2.16) i s used f o r these two reasons: i t does not require the inverse of [T ] and secondly, the program calculates [Y ] f i r s t anyhow, from which [ Y p k a s e ] i s obtained by inversion. [ Y m o d e ] i s then e a s i l y obtained by taking the r e c i p r o c a l of the diagonal elements of the r i g h t -hand side of equation (2.16). [ Z m ° d e ] i s not calculated from equation (2.15), but i n a simpler way from [ Z m ° d e ] = [A] [ Y m o d e ] - ^ (2.17) that i s , each component i s simply „mode l i Ymode ( 2 < l g ) This i s v a l i d because [A] from equation (2.11) can be rewritten as IA] = [ T ^ " 1 [ Z P h a s e ] [ Y P h a S e ] [ T J . = [ T , ] " 1 [ Z p h a S e ] [T.]. [ T . ] " 1 [ Y P h a S e ] [ T J = [ Z m o d e J . [ Y m ° d e ] (2.19) 3. Rotation of eigenvectors f o r zero shunt conductance It has to be noted that the eigenvectors (columns of [T^] or [T y]) are only determined to within a m u l t i p l i c a t i v e constant. Each eigenvector can, therefore, be m u l t i p l i e d with any non-zero complex s c a l a r , and i t w i l l s t i l l be the correct eigenvector. Since we assume zero phase shunt conductances (corona losses w i l l be discussed l a t e r i n Chapter 4), the modal conductances should also be zero. This can be achieved by multiplying the eigenvectors with a properly chosen constant. Then equation (2.8), which i s defined i n the frequency domain, 23 can be rewritten i n the time domain as follows 8.mode 1 3x = [ cm o d e ] 3 modd v 9t In order tp obtain zero modal conductances, a ro t a t i o n scheme i s used which makes the modal admittance matrix [Y m o c^ e] purely imaginary, mode = { } j [ B m ° d e ] ^ L J r o t a t e L J J L J r o t a t e This r o t a t i o n i s equivalent to d i v i d i n g the i - t h eigenvector ( i - t h column of [T ]) by a factor D^. F i r s t , f i n d the angle 6^ of Y ^ m o d e , as shown in Figure 2.1. Then 90° - 0. D. = e I (2.20) With a l l Dj^'s forming a diagonal matrix [d], the modified matrix of eigenvectors becomes IT.] = [T.] [D] 1 i J r o t a t e • i J L J (2.21) Then from equation (2.15a), [ Y m ° d e ] r o t a t e = [D] [ T . ] " 1 [ Y P h a S e ] ( [ T . ] 1 ) " 1 [D] (2.22) or [ Y m ° d e ] r o t a t e = [D] [ Y m ° d e ] [D] (2.23) Since a l l matrices i n equation (2.23) are diagonal, equation (2.23) i s simply a to t a t i o n of Y m ° d e by an angle (90° - 6^, which according to „ , , r„mode, , Figure 2.1, makes [Y ^rotate P u r e l y imaginary. 24 After [ Y m ° d e 0 i s found from equation (2.23), and [T.] 1 rotate H ' 1 i rotate from equation (2.21), [ Z m ° d e ] i s calculated from n rotate mode [ Ymode - 1 rotate rotate These modal quantities and transformation matrices obtained are c h a r a c t e r i s t i c s of the p a r t i c u l a r design of the untransposed l i n e . These modal parameters and modal transformation matrices are needed as input for the representation of untransposed d i s t r i b u t e d - parameter l i n e s in the time domain s o l u t i o n , such as i n the UBC version of the E l e c t r o -13 14 magnetic Transients Program as described i n ' Figure 2.1: Complex Y m o d e before and a f t e r r o t a t i o n 25 4. Confirmation of accuracy of eigenvalue and eigenvector subroutine The UBC Computing Centre l i b r a r y subroutine DCEIGN^^ i s chosen to compute the eigenvalues and engenvectors of the [Y]-[Z] matrix. This double p r e c i s i o n subroutine f i r s t reduces the complex matrix to a Hessenburg matrix H. The subdiagonal elements of H are then forced to converge to 49 zero by the modified LR method. Hence the diagonal elements of H converge to the eigenvalues. The eigenvectors can then be obtained by backward su b s t i t u t i o n . The correctness of the program has been checked by comparing i t s output with published r e s u l t s f o r a doub l e - c i r c u i t line"*"^. Both r e s u l t s of modal attenuations and modal v e l o c i t i e s agree to within three d i g i t s (see Table 2.1). The modal matrices [T ] d i f f e r only s l i g h t l y (see Table 2.2) . Table 2.1 UBC & BPA modal analysis r e s u l t s for a 735 kV l i n e 16 Modal attenuation .15998E6 .18438E6 .18497E6 .18606E6 .18615E6 .18614E6 neper/mile BPA .15998E6 .18437E6 .18497E6 .18605E6 .18614E6 .18614E6 Modal v e l o c i t y UBC .61227E-1 .19050E-2 .18209E-2 .54529E-3 .50169E-3 .47704E-3 miles c/s BPA .612E-1 .191E-2 .182 E-2 .544E-3 .502E-2 .475E-3 5. Real-valued frequency - independent transformation matrix Time domain solutions with the transformation matrix [T_^] become d i f f i c u l t i n theory since [T^] i s complex as well as frequency- dependent. Table 2.2 UBC and BPA modal matrix [T ] r e s u l t s for a 735 kV l i n e . 3412-j.0022 .3948-j.0157 ,4822-j.0 . 3412-j.0022 .3948-j.0157 .4822-j.0 . 3412-j.0022 .3955-J.0157 .4832-j.0 .3412-j.0022 .3955-j.0157 .4832-j.0 .5558+j.0 •3324+j.0230 -.3128+J.0248 .5558+j.0 .3324+j.0230 -.3128+J.0248 •5558+j.0612 .3294+J.0594 -.3153-j.0095 •5558+j.0612 .3294+j.0594 -.3153-j.0095 -.4959-j.0262 .5486-j.0 -.1118+J.0056 -.4959-j.0262 .5486-j.0 -.1118+j.0056 -.4959+j.0452 .5453-j.0784 -.1105+j.0215 -.4959+j.0452 .5453-j.0784 -.1105+j.0215 .1730-j.0017 .4647-j.0247 .5410-j.0 -.1730+J.0017 -. 4647+j.0247 -,5410+j.0 .1730-j.0298 .4681-J.0598 .5469-j.0408 -.1730+J.0298 -.4681+j.0598 -.5469+j.0408 .3209+j.0008 .4804+j.O -.4145+j.0304 -.3209-j.0008 -,4804+j.0 .4145-J.0304 .3209+j.0208 .4659+j.0285 -.4064+J.0048 -.3209-j.0208 -.4659-j.0285 .4064-j.0048 .6827+j.0 -.3550-j.0021 •0812+j.0061 -.6827+j.0 .3550+j.0021 -.0812-j.0061 .6827+j.0737 -.3637-j.0422 .0869+j.0159 -.6827+j.0737 •3637-j.0422 -.0869+j.0159 27 However, the imaginary part of the matrix [T ] i s always small 5%) compared with i t s r e a l part. By taking the r e a l part or the magnitude of the matrix i t s e l f , we obtain modal parameters which-are • s t i l l accurate enough ( ^ 2% deviation). Furthermore, the attenuation caused by corona may be much higher than that caused by the series resistance and for close-by strokes, trans-mission l i n e s should be represented as l o s s l e s s . With the approximations, the frequency dependence of the modal transformation matrix disappears. It i s therefore recommended that the complex matrix be approximated by a real-valued, frequency-independent matrix. This makes simulations much easier f o r two reasons: a) A frequency independent modal matrix does not require recompu-ta t i o n of the modal matrix at each frequency considered within the l i g h t n i n g frequency range, e.g. 10 kHz to 100 kHz. b) A real-valued modal matrix enables d i r e c t transient simulation to be performed i n the time domain. 6. Frequency dependent e f f e c t s i n l i g h t n i n g surge propagation To include frequency dependent effects i n transient overvoltage 17 18 studies i s a complicated topic by i t s e l f . Meyer, Dommel and Marti have investigated the t ime domain methods using convolution i n t e g r a l s and weighting functions. However, the frequency domain solutions can also be Q obtained by the Fourier Transformation methods. Though the frequency domain method i s inadequate to account for the non-linear phenomenon (e.g. corona discharge) and the time domain phenomena (e.g. i n s u l a t o r back — flashover or a r r e s t e r operation), i t i s s u f f i c i e n t for the purpose 28 of studying frequency dependent e f f e c t s on l i g h t n i n g surge propagation i n over-head l i n e s . 9 As discussed i n an e a r l i e r work , the frequency domain solutions includes frequency dependence of l i n e parameters. It also uses the exact complex frequency dependent transformation matrix to be computed at each frequency point, and employs the l i n e a r i n t e r p o l a t i o n technique i n evaluating the Fourier Transformation i n t e g r a l s . The r e s u l t s from a measured f i e l d test 19 20 by Ametani ' of a laboratory generated distant l i g h t n i n g wavefront 83.212 km from the substation was s u c c e s s f u l l y duplicated by the author using the Fourier Transformation method. (See Figure 2.2). Due to the frequency de-pendent e f f e c t of the l i n e parameters, an i n i t i a l r i s e time of 2 ys of the wavefront now increased to about 40 ys as the wave t r a v e l l e d down the l i n e . Thus, the frequency dependent e f f e c t must be included for th§ distant l i g h t n i n g stroke case. The l i g h t n i n g waveshape obtained a f t e r the stroke has t r a v e l l e d from the s t r i k i n g point to the substation can then be interfaced with the time domain solutions as used i n an e l e c t r o -47 magnetic transients program. For close-by l i g h t n i n g strokes, the r e s u l t i n g waveshapes can again be obtained by the Fourier Transformation integrals, and the simpler time domain methods. For the time domain method,- the multi-phase untransposed l i n e can be f i r s t solved by modal analysis using frequency independent parameters and real-valued transformation matrix (as described in previous s e c t i o n s ) . Then, t h i s multi-phase l i n e i s represented by a single phase l i n e approximation. As shown in Figure 2.3, r e s u l t s obtained by a l l these methods agree quite well (< 4% deviation). The single phase l i n e representation with frequency independent e f f e c t i s v a l i d i n t h i s c l o s e -by stroke case because v a r i a t i o n s among the modal a r r i v a l times at range of l i g h t n i n g frequencies are not apparent i n such short distances (e.g. < 2 km). Voltages(p.u) 3<|> with frequency dependence 3<j> without frequency dependence without frequency dependence ime Figure 2.3: Close-by l i g h t n i n g stroke case solved by frequency and time domain methods. 30 t=0 83.212 km A B • C Output voltage (p.u.) 1.0 0.8 0.6 •0.4 0.2 -0.2 f i e l d measurements 3<j> with frequency dependence Figure 2.2: Numerical simulation of over-voltage taking untransposition and frequency dependence into account.(Ref. 9) 31 In such a representation, s e l f impedance - parameters calculated at higher frequencies (e.g. 1 MHz) should be used to approximate the frequency dependance c h a r a c t e r i s t i c s of the l i n e . However, caution must be taken in choosing l i n e r esistance for the l i g h t n i n g surge studies. The frequency dependence of l i n e parameters of one phase for a t y p i c a l 500 kV l i n e i s shown i n Table 2.3. I t i s shown that the attenuation of the wave i s n e g l i g i b l e (< 5%) up to about 100 kHz for 1 km. The resistance to reactance r a t i o i s also small e s p e c i a l l y at higher frequencies, e.g. 2.8% at 1 M Hz. Furthermore, since the Bergeron's method of c h a r a c t e r i s t i c i n solving the transmission l i n e equation i s v a l i d only for a l o s s l e s s transmission l i n e , d i s t r i b u t e d l i n e losses are usually approximated by lumping the resistance at c e r t a i n l o c a t i o n s . This high resistance at 1 M Hz may cause inaccuracy i n the simulation. On the other hand, surge impedances calculated by zsurge = A + J^L ( 2 > 2 5 ) where R/jwL = 2.8% at 1 M Hz (See Table 2.3) zsurge = fcjL ( 2 > 2 6 ) / JUJC are e s s e n t i a l l y i d e n t i c a l for t h i s lossy and l o s s l e s s cases. Thus, com-p l i c a t e d frequency dependent e f f e c t s for the nearby stroke case can be ignored, and . frequency independent and l o s s l e s s representation give acceptable accuracy(.,,See Figure 2.9 ). ./ . Therefore,.the previous methods of modelling l i n e losses by simple • -i j • 1 2 • i 11,21 exponential decay i n overvoltages or any resistance lumping scheme are not acceptable. They should be replaced by d e t a i l e d weighting function techniques, or Fourier Transformation methods for distant stroke, or by 32 Frequency Resistance Reactance R/X (Hz) R(fi/km) X(ft/km) % zsurge v e l o c i t y Attenuation ( a ) (m/ys) e" Y^( /km) i n 6 183. 6525. 2.8 291. 280. .73 i o 5 42. 692. 6. 300. 272. .93 i o 4 7. 78. 9. 317. 257. .989 i o 3 .9 8.9 10. 340. 240. .999 Table 2.3: Frequency dependence of s e l f quantities of l i n e parameters for a 3 phase 500 kV l i n e . 33 l o s s l e s s l i n e representation f o r a nearby stroke as described i n above, 7. Determination of the surge impedance of the struck phase of a transmission l i n e . An accurate and r e l i a b l e value of the surge impedance in phase domain must be obtained as due to the following reasons: a) The amount of overvoltage wave transmitted from the overhead l i n e to the underground SF^ cable at the c a b l e - l i n e junction i s determined by the surge impedances of d i f f e r e n t components. The r e f r a c t i o n c o e f f i c i e n t t C _ i s R cable" C R , 2 Zsurge 7 ) zl:i.ne + ^cable surge surge where ^cable _ g u e i m p e d a n c e of cable (e.g. 60 ft) surge z l i n e _ g u r impedance of l i n e (e.g. 304 ft) surge b) The exact value of the overvoltage wave on the li n e , resulting from the l i g h t n i n g stroke (1^) i s d i r e c t l y r e l a t e d to surge -i . r,line impedance of the lxne Z as r surge v = ^ . z l i n e (2.28) 2 surge This r e s u l t i n g overvoltage wave impresses e l e c t r i c a l stress on external and i n t e r n a l i n s u l a t i o n of the system and forms the main concern i n the i n s u l a t i o n co-ordination study. 34 im In s p i t e of the above important c r i t e r i a , u n c e r t a i n t i e s i n surge pedance ca l c u l a t i o n s of overhead l i n e do e x i s t . ^ ' ^ ' ^ Reference 11 give r e l a t i v e l y lower surge impedance r e s u l t s f o r the ground wire (352 ft) 12 as compared to Darveniza's computation. Darveniza claims that the equation for surge impedance i n phase domain as i s given by: Z S U * f = 60 In ^ " (2.29) s e l f r Z S U r g € \ = 60 In ^- (2.30) mutual b.. where h = conductor height ; r = conductor radius a. .. = separation between conductors i j b. . = separation between conductor and i j other conductor image This i s r e a d i l y derived from the p o t e n t i a l c o e f f i c i e n t P and the inductance term L as: 5 s e i f • h r r ^ r - " l n T < 2- 3 I> J s e l f L 1 = Ho- to ^h ( 2 - 3 2 ) s e l f 2TT r and Z ^ f = / ^ - - - A _ _ P = 60 £ n ^ (2.33) s e l f / C ... s e l f s e l f ~ s e l f 35 where u = permeability ° -7 = 4TT x 10 H/m e = permit i v i t y ° 1 -9 = x 10 F/m 36TT [C] = [ P ] _ 1 [P] = p o t e n t i a l c o e f f i c i e n t matrix, with diagonal term P s e l f [L] = inductance matrix, with diagonal term L ° s e l f However, the above formulae neglect • Carson's correction terms, other conductors, and ground wires used f o r earth return. A d e t a i l e d c a l c u l a t i o n for the surge impedance matrix i n phase domain [Z ] & r v surge must be performed to in order to j u s t i f y t h i s assumption. If we consider the r e l a t i o n s h i p between the surge impedance matrix in both phase and modal domain as [ y p h a s e ] = [ z p h a s e ] [ ; I p h a s e ] ( 2 _ 3 4 ) and [ V m ° d e ] = [ Z m ° d e ][ I m ° d e ] (2.35) surge then by s u b s t i t u t i n g eqs.(2.3) & (2.4) into (2.35), we can get [T TV1**36] = [ Z m ° d e ][T ] _ 1 [ l p h a S e ] (2.36) v surge I or [ V p h a S e ] = [T ] [ Z m ° d e ] [ T . ] - 1 [ l p h a S e ] (2.37) v surge I 36 Comparing equations (34) and (37) , we thus obtain [Z phase surge ] = [T ][Z .mode 'surge J[T.] -1 (2.38) The above r e l a t i o n i n equation (2.38) i s i d e n t i c a l to that derived by Wedepohl. 22 In h i s method, the r e f l e c t i o n c o e f f i c i e n t f o r phase current i s f i r s t obtained. The c o e f f i c i e n t i s then set to zero to obtain the expression f o r [ Z P ^ a s e ] as in equation (2.38). surge M Results for the surge impedance from equations (2.29) and (2.38) for both the ground and the phase wires are shown in Table 2.4. As can be seen from the table , the surge impedance obtained by Darveniza's formula which neglects the skin e f f e c t of the earth return component introduces n e g l i g i b l e deviation (about 1%). However, the Darveniza's formula should only be used when the ground wire i s treated as another i n d i v i d u a l phase (e.g. for the close-by l i g h t n i n g stroke case). If one takes the ground wire as another component for earth return (e.g. for the distant stroke case), the formula for s e l f surge impedance must be modified accordingly by t r e a t i n g voltages on ground wire to be zero. This requires reducing the impedance and admittance matrices before surge impedances can be calculated. The surge impedance value obtained i n t h i s case i s lower than that obtained by Equation (2.29), as shown in Table 2.4. 8. Single phase representation for close-by strokes on double c i r c u i t e d l i n e A f t e r the author has v e r i f i e d that s i n g l e phase representation with appropriate choice of l i n e parameters i s accurate for a three phase l i n e case without ground wire r a double-circuited overhead transmission l i n e 23 of the MICA Project of the B.C. Hydro and Power Authority was used as a more de t a i l e d transmission system with ground wires. 37 surge Impedances ground wire phase wire A 547 ft 342 ft B or C 545 ft 338 ft D - 318 ft (A-B)/A*100% 0.4% 1.2% A = Exact method (2.38) with Carson.':s Correction terms for earth return skin e f f e c t , ground wire treated as another phase. B = Exact method equation (2.38) without Carson's Correction terms f o r earth return skin e f f e c t . C = Darveniza approximate equation (2.29). D = Exact method equation (2.38) with Carson's Correction terms for earth return skin e f f e c t , ground wire treated as earth return component. Table 2.4: Self surge impedances for ground and ^ 7 phase wires f o r a t y p i c a l 500 kV l i n e ' 38 This tranmission system i s a double-circuited 500 kV l i n e . Each tower consists of a three phase l i n e with two ground wires (See Figures 2 .4 and 2.5). When the l i g h t n i n g stroke h i t s e i t h e r one of the ground wires or one of the phase conductors, d i f f e r e n t l i n e parameters must be chosen because of d i f f e r e n t l i n e design. The corresponding parameters are shown in Table 2.5. One can see from t h i s table that the s e l f surge impedance of the ground wire i s greater than that of the phase conductor. The wave propagation v e l o c i t y i s also lower i n the ground wire case. In Figures 2.6 and 2.7, one can compare the l i g h t n i n g overvoltage wave propagation c h a r a c t e r i s t i c s f or the open and short c i r c u i t t e s t by using m u l t i - and single-phase l i n e representation when l i g h t n i n g stroke, h i t s the ground wire. Figure 2.6a shows the r e s u l t obtained by the m u l t i -phase s o l u t i o n method using modal a n a l y s i s . I t also shows c l e a r l y the d i f f e r e n t modal components on the r e s u l t i n g waveform. Figure 2.6b shows the r e s u l t f o r the si n g l e phase case and the o v e r a l l important propagation c h a r a c t e r t i s t i c s of multi-phase representation i s s u c c e s s f u l l y duplicated here. Similar r e s u l t s are obtained for the s h o r t - c i r c u i t t e s t , as shown i n Figures 2.7a and 2.7b. One can observe that the current waveforms obtained from these two d i f f e r e n t l i n e representations agree very w e l l . S i m i l a r l y , the open and short c i r c u i t t e s t r e s u l t s are also s u c c e s s f u l l y duplicated for surges on the phase conductor as i n c a s e of d i r e c t s t r o k e s or b a c k f l a s h o v e r s ( See Fig.2.8&2.9) Thus, i t i s recommended to use si n g l e phase representation for double-circuited l i n e with ground wires for studying close-by l i g h t n i n g stroke propagations. 23 unit 5-3/4x10" shielding ground tower insulator //wires extending strings / / 1.6km beyond substn 3 phase conduc tors tower Ty p i c a l l i g h t n i n g arrester c h a r a c t e r i s t i c s : nominal rating (reseal voltage), switching sparkover Min 60 Hz sparkover lightning sparkover 3 phase underground SF- cables 6 -l i g h t n i n g a r r e s t e r Jat transformer ^jO^L-^ cable junction .transformer to be protected 420 kV 950 kV 568 kV 985 kV 396 kV 960 kV-555 kV 990 kV-Insulation levels 60 Hz BIL SF, cable Transformer 6 800 kV 745 kV 1550 kV 1675 kV Figure 2.4: Layout of SFg substation protection scheme showing one of the double c i r c u i t systems. 40 10 — 9 4 1 5 8 7 6 1 2 3 Conductors 1-3, 6-8 phase wires 4,5, 9,10 ground wires Coupling f a c t o r : l i g h t n i n g struck <j) - wire K 34 ^phase surge 3,4 ^phase surge 3,3 •16 y induced ground wire V phase wire l i g h t n i n g struck g - wire K ^phase _ surge 4,3 _ 43 phase Jsurge 4,4 •06 induced phase wire V ground wire Figure 2.5: Side view of the MICA 10 <j> Systems. Ground conductor Phase conductor Self surge impedance Z ^ | e 658 ft 304 ft Wave v e l o c i t y v 245 ™/\is 293 m / y s Line resistance 0 /^m 0 /^m Length 1609 m 1609 m where Z S ^ f = 60 In ^ s e l f r = /L P 77 s e l f s e l f v e l o c i t y = s e l f L s e l f and L . , _. and P . , , are diagonal elements of s e l f S 21 matrix [L] and [P]. Table 2.4: Line parameters of ground and phase conductor for l o s s l e s s single phase representation. 42 Dif f e r e n t modal a r r i v a l times Single phase a r r i v a l In 1<|> case: V T 2 p 3 = (Coupling factor) surge = 3 ' 4 V surge T2G4 Z4,4 Figure 2.6: Open c i r c u i t t e s t on s i n g l e and multi-phase representation with stroke on ground wire. 4 3 i D i f f e r e n t modal a r r i v a l times Figure 2.7: Short c i r c u i t t e s t on s i n g l e and m u l t i -phase representation with stroke on ground wire. Figure 2.8: Open c i r c u i t t e s t on sin g l e and multi-phase representation with stroke on phase conductors. 45 Small d i f f e r e n c e i n d i f f e r e n t modal a r r i v a l times _3 iT 2 p 3 ( 1 0 P-u.) t-0- \. 304« TP3 T2P3 1+ 4 2 2 p.u. (S T2P3 Time t} ii,.or 4 f 8 12- 16 (ys) Single phase a r r i v a l Figure 2.9: Short c i r c u i t t e s t on si n g l e and multi-phase representation with stroke on phase conductors. 46 CHAPTER 3: LIGHTNING WAVE PROPAGATION IN SF 6 GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM. 1. Introduction The world's f i r s t commercial SF, gas - i n s u l a t e d cable rated at 6 345 kV was i n s t a l l e d i n 1970. I t s inherent advantages over conventional underground o i l - f i l l e d cables with respect to charging current, d i e l e c t r i c losses, thermal performance, voltage r a t i n g f l e x i b i l i t y and power handling capacity are well-known. It o f f e r s a d d i t i o n a l advantages of reduced substation s i z e . This compactness i n si z e of SF^-insulated substations and switchgear brings the equipment closer to the protective l i g h t n i n g arrester located at the overhead l i n e and underground cable junction. This i s of.vital-importance, e s p e c i a l l y when there i s no l i g h t n i n g a rrester at the transformer terminal, as in c e r t a i n substation design. 23 In the SF,-insulated cable at the MICA Dam, which w i l l be used o as a test example, each of the 3 phase cables consists of two concentric aluminum tubes (see Figure 3.1a). The inner tube i s the conductor core and the outer grounded tube i s the sheath. The three sheaths are s o l i d l y bonded together and grounded at many lo c a t i o n s . At the high frequencies encountered i n l i g h t n i n g surges, the sheath return current w i l l be equal i n magnitude and 180° out of phase with the core conductor current. Whether the magnetic f i e l d external to the sheaths can be completely neglected in the frequency range of in t e r e s t must be investigated, however. I f the magnetic f i e l d i s n e g l i g i b l e , then there would be no mutual inductive coupling among the phases. There i s no e l e c t r o s t a t i c c apacitive coupling between phases as the solidly-grounded sheaths act as e l e c t r o s t a t i c shields U p to 1 Mhz. 47 Skin depth o f A l , •t * \-1/2 8.1 = ( . T T f a u ) = cm =1.0 cm at 60Hz Scale--= 1:2.54 r = r„ = 3" = 7.62 cm 3.5" = 8.89 cm 9.75" = 24.765 cm r. = 10" = 25.4 cm 4 sheath thickness=.635 cm Permeability A l = y u, = u-r 0 0 = 4TT .x 10 -7 H/ m Figure 3.1a: Individual cable design P e r m i t t i v i t y SF & = £ r £ Q = eQ 1 x 10 9 F/m 36TT Figure 3.1b: Overall cable layout. 48 In order to investigate the sheath current return phenomenon of SF, b cables, and thus to c l a r i f y the wave propagation c h a r a c t e r i s t i c s , the cable parameter must be calculated accurately. In t h i s research work, cable 24 25 parameters of multi-core cable ' i s not investigated. Such multi-core cable systems c e r t a i n l y have coupling between phases at a l l frequencies. Nevertheless, coupling between phases for the s i n g l e core cable system i n d i f f e r e n t frequencies needs further i n v e s t i g a t i o n . 26 27 Commellini and Abledu used f i n i t e elements technique to sub-divide the main conductors into smaller sub-conductors of c y l i n d r i c a l shape. The impedance matrix f o r the main conductors was formed by bundling up the sub-conductors i n the matrix elimination process. However, due to the t h i n tubular shape of the conductors involved i n the SF^ buses, large number of sub-conductors i s required. This w i l l demand huge computer core storage space and long computer execution time. 28 29 Sunde and Pollaczek had derived a n a l y t i c a l expressions for the s e l f and mutual impedance of cables which are constructed overhead, underground or on ground surface. These a n a l y t i c a l expressions contain Kelvin functions and an i n f i n i t e i n t e g r a l known as the Carson's Correction terms. Before the widespread a p p l i c a t i o n of d i g i t a l computers, s i m p l i f i e d assumptions and r e s t r i c t i o n s were made to f a c i l i t a t e the computation process. With the recent popularity and increased a p p l i c a t i o n of d i g i t a l computers, these i n f i n i t e i n t e g r a l s can be modified and replaced by s t r a i g h t forward numerical computations without s i g n i f i c a n t s a c r i f i c e for accuracy. 30 31 32 Wedepohl et a l and Ametani ' used d i f f e r e n t approaches to tackle the a n a l y t i c a l expression i n the cable parameter c a l c u l a t i o n . However, both approaches gave d i f f e r e n t r e s u l t s (20% from each other). Bianchi proposed to ca l c u l a t e the earth or sea return impedance by approximating the return medium as a tube of i n f i n i t e outside radius. These approximation r e s u l t s f e l l somewhere between those of Wedepohl et a l and Ametani. Because of the inconsistency i n the above f i n d i n g s , a d e t a i l e d i n v e s t i g a t i o n f o r numerical c a l c u l a t i o n of cable parameters must be performed to reveal wave propagation c h a r a c t e r i s t i c s i n SF^ sing l e core cables. Current return c h a r a c t e r i s t i c s from core through sheath also must be investigated to confirm s i n g l e phase or multi-phase representation for the cable system involved. 2. Formation of series impedance matrix f o r SFfi cables The s i n g l e core SF^ cable system configuration i s shown in Figure 3. Each phase consists of two conductors, core and sheath. We can b u i l d up a 6 x 6 series impedance matrix Z, describing the cable system as follows: d V c l n dx z i i s l l Z 19 sl2 Zml2 Zml2 Zml3 Zml3 " ^ l " dV . s i dx Z 01 s21 Zs22 Zml2 Zml2 Zml3 Zml3 ^ 1 d V c 2 dx Zml2 Zml2 z i i s l l Z s l 2 3nl2 Zml2 ^ 2 dV s2 dx Zml2 Zml2 Zs21 Zs22 Zml2 Zml2^ • ^ 2 dV _ c3 dx Zml3 Zml3 Zml2 Zml2 Z s l l Z s l 2 ^ 3 dV _ s3 dx _ Zml3 Zml3 Zml2 Zml2 Zs21 Zs22 _ . ^ 3 . 50 z s Zml2 Z, " ml3 ^ 1 Zml2 Z s Zml2 \l ^ 2 (3.1) Zml3' T Zml2 Z s ^3 s where Z is... 'self submatrix on the diagonal. A l l Z matrices are s ° s equal because they represent i d e n t i c a l cable configurations. I t i s assumed: that the mutual impedance between cores, between sheaths and between corresponding cores and sheaths are a l l equal. In other words, a l l elements i n the sub-matrix Z , „ or Z. are ^assumed ml2 ml3 to,; be equall:J:-(Bee Section 6 f o r "further discuss ion) • — 3. C a l c u l a t i o n of s e l f and mutual earth return impedance f o r s i n g l e core cable The a n a l y t i c a l expressions f o r the s e l f and mutual earth return 29 impedance of cables was f i r s t derived by Sunde and Pollaczek^ and then by Wedepohl et a£? F i r s t l y , t h e Maxwell's electromagnetic equation can be solved a f t e r neglecting end e f f e c t s as: 51 V x E = -ja>y0 H (3.2) V x H = J + |2. = j ( i + (3.3) 3t a = J , as displacement current , can be neglected. Taking the c u r l of equation (3.2) and su b s t i t u t i n g into (3 3) give V x V x E = -jwy 0 V x H or V ( V E ) - V 2E = - j u y Q ( J ) Assuming cable separation >> cable radius, we have V 2E = jtoy 0 a(E +p i 6(x) <5(y + h)) Assuming cables are p a r a l l e l to ground surface and attenuation of voltage and current i s n e g l i g i b l e over distances comparable to cable separation, 2 2 we have 3 E n 3 E, + —T- = 0 » y > o. ( 3 - 4 > 2 2 3x 3y 2 2 3 E„ 3 E —j- + Y = m 2E 2 +p m2 i 6(x) 6(y + h) , 3x 3y y < 0. (3.5) where m:; = J j cjy P . . . 1 P • = earth r e s x s t i v x t y = — oj = angular frequency y = permeability 6 = Difac function E^,E 2 = e l e c t r i c f i e l d above and below ground. Imposing the r e s t r i c t i o n of continuity of e l e c t r i c f i e l d at y = 0 for E^ and as boundary conditions, we can obtain a general expression 52 y 7 T T Figure 3.2: Conductor configurations for earth return formula. 53 for the underground e l e c t r i c f i e l d . We can then obtain the expression for the mutual earth-return impedance of underground cables by d i v i d i n g E 2(x,h 2) by the current i (O.h^) t o 8 e t Z. . = i j 2IT 0 0 / 2" 2* 0 0 '/2 2~\ exp(-£/ a +m ) & j ax ^ + j exp;(-l//a +m ) da -1 1 ' n~~2 a + Va +m 0/2.2 2/a +m exp( -£Va 2+m 2) 2v42+m2 da (3.6) exp,(= v^/a2-rm2: lax 2^.00 1 a I +/a2+ra2 da + (K n(mDj - (K 0(mD 0)) 2 77 s ( T 1 0: 2' = A Z 4 J + 4 ^ fe-JuDj - K„(mD„)) J i j 1 2 7 7 ^ o ^ l CT 2y (3.7) where = Kelvin function of order zero m D, = = y ^ ^ " j x = h o r i z o n t a l separation between cables /x /2 2 D 2 = /x + ( ^ 4 - ^ ) h^,h 2 = depths of b u r i a l of cables a = l = |h 1 + h I AZ.. = Carson's Correction term, i d e n t i c a l to that f o r over-head l i n e s case. The above formula i s also applicable to s e l f earth return impedance components. In such cases, the terms f o r and can be re-defined as D 1 = r D 2 = 2h (3.7a) • where r i s radius, h = depth of buried cable. However, the above formula i s unsuitable f o r st r a i g h t forward c a l c u l a t i o n s . Before d i g i t a l computers are widely used, approximate r e s u l t s were obtained 28 only a f t e r c e r t a i n l i m i t i n g conditions were accommodated. Then, Wedepohl et a l used Cauchy's integration for the Carson's correction terms and derived approximate formula f o r the equation given i n equation (3.7) as mD „ Z. . = ^ { £n(Y-±-) + i - f mA } (3.8) i j • 2 ir 2 2 i for |mD | < 0.25 and Y = Euler;'s constant = 0.5772157. However, the above formula gives r e s u l t s which are about 20% higher when compared with the d i r e c t numerical computation using the o r i g i n a l equation as i n Ametarii's case. The i n f i n i t e i n t e g r a l for the underground cable i s the same as the Carson's correction terms for overhead l i n e s . We can define a new parameter a as a = / ^ D = 4TT /5 10 4 D / | - (3.9) 55 where D and p are in MKS units 2h. for s e l f earth return impedance ^ ~ for mutual earth return impedance This correction term i n t e g r a l can be represented by the following . .„ . . 3 4 i n f i n i t e converging s e r i e s : 1. For a < 5 AR' = 4o)-10~4{^-o -b^a* cost)) 2 2 +b^[ (c2~lna)a cos2<{)+(j)a sin2cj)] 3 +b^a cos3<}> -d^a4cos4cf> -b^a^cos5<j> 6 6 +b^ [ (c^-lna)a cos6<j)+cba sin6c))] +b^a^cos7<j> g -d Qa cos8<f> o - ...} AX' = 4to-10 4{|<0.6159315-lna) +b^a*cos<j> 2 -d^a cos2cf> 3 +b^a cos3<j> 4 4 -b^[(c^-lna)a cos4cf>+(j>a sin4cf>] +b^a^cos5cj) -d^a^cos6tj) +b^a cos7<j> 8 8 -b Q[ ( c Q - l n a ) a cos8<J>+c(>a sin8c()] o o + ...} 56 Notice that each 4 successive terms form a r e p e t i t i v e pattern. The co-e f f i c i e n t s b., c. and d. are obtained from the recursive formulas: 1 1 l /2 ^ b^ = — - r f o r odd subs c r i p t s , b. = b. „ . z^foN with the s t a r t i n g value -\ l i-2 i(i+2) 1 ^•b„ = f ° r even subscripts, 2 I D c = c. „ + + -rrr with the s t a r t i n g value c„ = 1.3659315, l i-2 l i+2 - 2 A 7 1 U d. = j- • b., l 4 l with sign = ±1 changing a f t e r each 4 successive terms (sign = ±1 for i 1,2,3,4; sign = -1 for i = 5,6,7,8 e t c . ) . 2. For a > 5 , _ ' cos((> /2 cos2<)) cos3<}> 3cos5<j) 45cos7(j) - ( 2 3 5 " 7 ) 4a)-10 4 /I Q3.ll) -4 AX1 - ( C O S 1 ^ _ cos3<l> + 3cos5<{> + 45cos7(}> ^ # 4ai* 10 3 5 7 >=-a a a a v2 It should be noted that the correction terms w i l l become zero when the parameter a i s very b i g , i e . when frequency or cable distance from g.round i s very large or when earth r e s i s t i v i t y i s very small. The Kelvin functions can also be calculated by another i n f i n i t e 36 converging series as can be obtained from availabe source. , ''-It can also 35 be calculated by a s p e c i a l subroutine CBESK ava i l a b l e from the UBC Computing Centre. A f t e r e s t a b l i s h i n g the numerical formula for cable earth return impedance, the discrepancies between the r e s u l t s of Wedepohl et a l and those of Ametani can then be c l a r i f i e d . The author has confirmed that accurate cable earth return impedance •can be calculated by using d i r e c t computation of i n f i n i t e series sub-40 s t i t u t i o n f o r i n f i n i t e i n t e g r a l and Kelvin function. Cable mutual impedance from Ametanis computation i s acceptable though d i f f e r e n t i n f i n i t e s e r i e s i s used for the Carson's correction terms. I d e n t i c a l r e s u l t s are obtained at l e a s t to 4 s i g n i f i c a n t figures for frequencies up to 100 k Hz (See Figure 3.2). The approximate formula given by equation (3.8) on the other hand, gives r e s u l t s about 20% c o n s i s t e n t l y higher. The author has also confirmed that the earth return impedance for underground cable can be approximated by the equivalent earth return impedance for overhead l i n e s . The expression for earth return impedance for overhead l i n e i s D Z.. = to -==- + AZ.. , .„ i j 2TT DJ^ I J ' (3.12) for mutual earth return impedance 2h and Z. . = to — + AZ. . , (3.13) xj 2 It GMR xj for s e l f earth return impedance where i s distance between i * " * 1 and image of j 1 " ^ conductor; i s distance between i * " * 1 and j t ' 1 conductor; • h i s height of conductor above ground; GMR i s geometric mean radius= radius of conductor at high f r AZ i s Carson's correction term. I d e n t i c a l r e s u l t s up to 3 or 4 figures are obtained for earth r e s i s t i v i t y of 1 to 100 Q-m up to the frequency of 100 K Hz. The 58 consistency between these two r e s u l t s i s due to the fact that the Kelvin functions K (/j" x) = ker(x) + j kei(x).; 36 can be evaluated by the following convev^ing series for 0 < x < 8: ker x = — In ( i r ) berz+Jxbei x—.57721 566 1 -59.05819 744(jf/8)*+171.36272 133(x/8)8 : -60.60977 451(z/8)12+5.65539 121 (x/8)'8 -.19636 347(x/8)20+.00309 699(:r/8)24 -.00002 458(2-/8)28+« (3.14) H<ixio- 8 kei*=-ln($x)beia;-fir ber a,+6.76454 936(z/8)* -142.91827 687(i/8)6+124.23569 Q50(z/8y° -21.30060 904(ar/8)14+l. 17509 064(r/8)18 -.02695 875(ar/8)22+.00029 532(x/8)29+« _(3.15) |€|<3X10-» where ber x=1 -64(as/8)4+113.77777 774(*/8)8 -32.36345 652(z/8)12+2.64191 397(a;/8)16 -.08349 609(*/8)M+.00122 552(x/8)24 -.00000 90l(x/8)28+e |«|<ixio-» (3.16) bei x=16(a;/8)2-113.77777 774(x/8)* + 72.81777 742(a/8)l0~ 10.56765 779(x/8)u + .52185 615(z/8)l8-.01l03 667(ay8)22 +.00011 346(z/8)29+e M<8X10-» Figure 3.2: Mutual impedance between outermost cables by d i f f e r e n t computation methods. 60 and x = ^f- D, (3.18) u = 4TT x 10 ^ ^m oi = frequency D = distance Di o r D L 2 P = earth r e s i s t i v i t y For frequencies up to about 1 M Hz and earth r e s i s t i v i t y of about 100 ft-m,and cable separation or cable depth of about 1 m, the term x i s r e l a t i v e l y small as x -> 0 Then, one can rewrite equations (3.14) to (3.17) for x 0 as ber = 1 b e i = 0 • • K Q(/fx) = ker x + j k e i x = - £n |- x - 0.57721 - j J (3.19) Thus, for the earth return impedance as shown i n equation (7), we have Z. . = AZ. . + ™ (K (mD.) - K (mD.)) xj xj 2 IT o 1 o 2 mDn - 4 Z« + if « - «• -r - - jj) -mD ( - in -f - - .57721 - jJ-)) = AZ . + ^ • in ^ , x * 0 y 1 J 2 l T j , °1 Carson's correction term self-term which i s the same as i n equations (3.12) and (3.13). 61 The numerical r e s u l t s f or the self-term component of the s e l f and mutual earth return impedance for the underground cable obtained by the d i f f e r e n t formulae developed e a r l i e r are shown in Table 3.1. As can be seen from Table 3.1, the r e s u l t s obtained by these d i f f e r e n t formulae are very consistent. The f i n a l r e s u l t s f or mutual impedance from these methods are also shown i n Figure 3.3 for frequencies up to 1 M Hz. Because of the observed consistencies, the overhead l i n e formula approximation i s therefore recommended f o r underground cable for a l l frequencies up to 1 M Hz and earth r e s i s t i v i t i e s above 1 ft-m. 4. Calculation of s e l f impedance matrix for single core cable After the s e l f and mutual impedance for earth return of under^. ground cables,.is obtained, the s e l f impedance of i n d i v i d u a l cables can be calculated and the obtained r e s u l t s f or d i f f e r e n t current loops can then be transformed to the required form for the impedance diagonal submatrix Z as shown i n equation (3.1). s At f i r s t , one can consider the current in each of the i n d i v i d u a l cables flow i n two adjacent loops as shown in Figure 3.4. Loop 1 i s formed by the current flowing through the core and returning through the outside sheath. Loop 2 i s formed by the current flowing through the sheath and returning through the outside earth. These two loops can be described by equation (3.21) as d v l dx Z l l Z12 4 1 1 d v 2 dx _ Z21 Z22 _ i 2 _ where Z = Z by symmetry. 62 Separation i n meter D l D2 (1,3) .889 2.189 .901+j.OOO (60Hz) .901 .895-J.020 (100Hz.) .901 (1,2) 1.778 2.676 .408+j.OOO (60Hz) .408 .403-J.018 (100Hz) .408 (s e l f ) .254 2.0 2.064+j.OOO (60Hz) 2.064 2.057-J.022 (100Hz) 2.064 where f = 60 Hz, m = / j 6 ° X 2 \ l Q ^ X 1 0" 7-= .0022 /J i TT /icoy /jlOO x 2TT x 4TT X 10 ^ A Q n rr f = 100 Hz, m = / ^ = / J JOO -089 / j underground cable (exact) Z i j = T r f ( V m V ' K 0 ( m D 2 » + A Z i j overhead cable (approximation to above) hi " ^ *> 57 + * Z 13 • - - » * 0 Table 3.1: Mutual and s e l f earth return impedance terms as given by under-ground and overhead cable formula. underground overhead Figure 3.3: Approximations of mutual impedances between underground cables by Carson's formulae. Figure 3.4: Current loops inside SF^ cable f o r s e l f impedance c a l c u l a t i o n . 65 The matrix elements of equation (3.21) can be obtained by considering the i n d i v i d u a l current loop components making up the corresponding loops 1 and 2 as Z = Z + Z + Z 11 core-outside core/sheath i n s u l a t i o n sheath-inside (3.22) Z = Z + Z 22 sheath-outside earth-inside (3.23) z = z = —z 12 21 sheath-mutual (minus sign since i and ± i n d i f f e r e n t d i r e c t i o n ) . (N.B. Z ^ i s n e g l i g i b l e when sheath thickness » skin depth) where the i n d i v i d u a l elements are (3.24) (Zl) Z core-outside (Z2) Z core/sheath i n s u l a t i o n (Z3) Z sheath-inside (Z4) Z sheath-outside (Z5) Z earth-inside i n t e r n a l impedance of core with return through outside (sheath). impedance of SF, i n s u l a t i o n due to the o time varying magnetic f i e l d . i n t e r n a l impedance of sheath with return through ins i d e (core). i n t e r n a l impedance of sheath with return through outside (earth). s e l f earth return impedance, t h i s can be calculated by equations (3.7) & (3.7a), or can also be obtained by equation (3.25) with the approximation of i n f i n i t e 33 outside radius (Z6) Z sheath-mutual = mutual impedance of tubular sheath between loop 1 i n inner surface and loop 2 i n outer surface of sheath. 66 The i n d i v i d u a l s e l f and mutual impedance terms can be again obtained by solving the Maxwell's equations for the coaxial conductors as s i m i l a r to equations "£3.2)&(3 .3) # They are a function of frequency as derived by 37 28 Schkelkunoff and Sunde as . . , = (I n(mq) K, (mr) + K„(mq) I.,(mr)) tube-inside 2iTqp 0 1 0 1 (3.25) tube-outside = 2 ~ - (I Q(mr) KL(mq) + K Q(mr) I^mq)) (3.26) Jtube-mutual with 2Trqrp p = I 1(mr) ^(mq) - I (mq) K (mr) (3.27) (3.28) where and y q r m = angular frequency = 2i:f permeability = P ^ Q J y r = 1 for A l outside radius of tubular conductor inside radius of tubular conductor p = d.c. r e s i s t i v i t y V r i K0''K1 p Bessel functions Kelvin functions A f t e r obtaining the i n d i v i d u a l terms of the loop equation matrix as shown in equation (3.21), we can then obtain the diagonal sub-matrix elements by applying the following c i r c u i t conditions: V = V - v 1 c s V = V 2 s (3.30) 1 = 1 1 c (3.31) i» = i + i 2 c s (3.32) The i n d i v i d u a l loop equations of equation (3 21) then becomes dV dV -T9" + -J^ = ( Z n + z i , ) i + Z i o 1 dx dx 11 12 c 12 s dV and — = (Z + Z,_) i + Z„„ i dx 12 22 c 22 s (3.33) (3.34) Adding equation (3.33) to (3.34) gives dV dx ( Z11 + 2 Z12 + Z22> \ + ( Z12 + Z22) is ' ( 3 " 3 5 ) Thus, we can rewrite the s e l f sub-matrix Z as ' s Z 1 1 + 2 Z 1 2 + Z 2 2 Z 1 2 + Z 2 2 dV c dx dV" s dx Z12 + Z22 J22 (3.36) 5. /ffheath current refragn c h a r a c t e r i s t i c s for, usual earth As current flows along the core of the buried SF^ bus, a return path i s formed on i t s own sheath and possibly also on the surrounding s o i l and adjacent sheaths. Whether a l l the currents w i l l return through i t s own sheath depends s o l e l y on the frequencies involved. Due to the skin e f f e c t i n sheath material (Aluminum), a l l core current w i l l return through i t s own sheath for frequencies above 1 k Hz. In r e a l i t y , the SF^ cable i s l a i d on the ground surface (e.g. inside the lead shaft) or i s constructed above ground and grounded at c e r t a i n i n t e r v a l s (e.g. inside the substation). This cable l o c a t i o n even favor more current returning through the sheath than the ground as compared to buried'cable case. One can thus in v e s t i g a t e the l i m i t i n g case with the cable buried underground. This cable l o c a t i o n w i l l favour l e a s t 68 core current returning through i t s own sheath. In order to investigate the sheath current return c h a r a c t e r i s t i c s and therefore the mutual coupling between cables, one has to use the seri e s impedance matrix. One can consider cases i n which adjacent sheaths are e i t h e r included or excluded. a. Sheath current return c h a r a c t e r i s t i c s for single cable system For t h i s case, one only has to consider the s e l f diagonal submatrix of the series impedance matrix as dV dx dV dx J s l l J s l 2 J s l 2 Js22 (3.37) Since the sheaths of the three i n d i v i d u a l SF^ cables are s o l i d l y grounded at short j o i n t i n t e r v a l s or l a i d on earth surface, or buried inside .the earth, the sheath voltages can be considered to be zero for a l l p r a c t i c a l purposes. Then Equation (3.37) becomes 0 J s l 2 Z or s!2 's22 i + Z c s22 I -c At high frequencies as sheath mutual impedance i s n e g l i g i b l e ^ 3 3 ) when sheath thickness greater \ t h a n skin depth( See Appendix A.). Thus, neglecting the other two sheaths, the current return c h a r a c t e r i s t i c s of the SF,, cable through i t s own sheath from the core o can be calculated as i n Equation (3.38). The obtained r e s u l t s are shown i n Figure 3.4 .. For t h i s case, e s s e n t i a l l y a l l the current through the core w i l l return through i t s own sheath above the frequency of 10 Hz. 69 "?"sheath core z 0 sheath-mutual '''sheath """core earth _ single-cable system three-cable system: -current i n one core • ' -current i n three cores ® - i B l / i c l ® " i s 2 / l c 2 ® " 1 s 3 / l c 3 p = lOOft-m depth = 0 or .254m log frequency 1 — 10 100 l k 10k (Hz) Figure 3.4: Ratio of core current return through own sheath for single-and three-cable system. 70 In other words, the sheath acts as a perfect magnetic s h i e l d above the frequency of 10 Hz. Because of t h i s consideration, a l l SF^ cables are decoupled from one another and can be represented as 3 s i n g l e phase systems. I t i s also found that a change i n depth (1 m to .254 m) of cable does not change the current return c h a r a c t e r i s t i c s noticeably. b. Sheath return c h a r a c t e r i s t i c s f o r 3-cable system Since a l l the 3 sheaths of the SF^ bus are s o l i d l y grounded, the current w i l l return through a l l the three sheaths at lower frequencies (< 60 Hz). At higher frequencies, however, a l l the core current w i l l return through i t s own sheath because of the skin e f f e c t on the sheath. Here, again, one can conclude that the three SF^ cables are decoupled from one another. For t h i s case of 3-cable system, one can also consider the sheath voltages to be zero. One can substitute t h i s condition into equation (3.1) and obtain dV " c l dx -5 s i i Z s l 2 Zml2 Zml2 Zml3 Zml3 ^ 1 0 Z s l 2 Zs22 Zml2 Zml2 Zml3 Zml3 ^ 1 d V c 2 dx Zml2 Zml2 Z s l l Z s l 2 Zml2 Zml2 \2 0 Zml2 Zml2 Z s l 2 Zs22 Zml2 Zml2 is2 d V c 3 dx Z'ml3 Zml3 Zml2 Zml2 Z n s l l Z s l 2 c3 0 Zml3 Zml3 Zml2 Zml2 Z s l 2 Zs22 s3 (3.39) (N.B. Z',„ = Z as symmetrical arrangement of cables as ml2 mz 3 i n Figure ,3.1b) Equating the zero sheath voltages f or the 3 cables, we have 0 = Z c 1 0 i . + Z „ i , + Z. . . ( i + i ) + Z (1 + i ) (3.40) sl2 c l s22 s i ml 2 c2 s2 ml 3 c3 s3 0 = Z 1 0 ( i . - . + i ) + Z i . + Z i + Z 1 0 ( 1 . + i ) (3.41) ml 2 c l s i s l 2 c2 s22 s2 ml2 c 3 s3 0 = Z , 0 ( i - + i n) + Z. , 0 ( i „ + i 0) + Z _ i _ + Z 0 „ i _ ,~ / 0 . ml3 c l s i ml 2 c2 s2 s l 2 c 3 s22 s3 (3.42) If we assume phase B i s energized, i . e . , we assume ± c l = i c 3 (3.43) ^ 1 = S 3 ( 3 - 4 4 ) i t l = i c 3 = 0 (3.45) Then, substitute equations (3.43) to (3.45) into (3.40), we obtain 0 = Z c 0 0 i + Z i + Z i + Z i S 2 2 s i ml2 s2 ml3 s i m!2 c2 = (Z. ,n_ + Z ) i + Z. i + Z i - ml 3 s22 s i ml2 c2 ml2 s 2 Znil3 + Zs22 X s l , 1s2 , / 0 or + = -1 (3.46 ml2 1 c 2 1 c 2 Also substitute equations (3.43) tO (3.45) into (3.41), we obtain 0 " 2 Zml2 \l + Zs22 \2 + Z s l 2 ! ! i ^ . i 5 l + ! 5 2 2 ^ = _ ± (3.47) Z s l 2 Xc2 Z s l 2 1 c 2 72 By solving equations (3.46) and (3.47), we get Zml3 + Zs22 ^ml2 2Z ml2 J s l 2 Zml3 + Zs22 Jml2 2Z ai2 'sl2 -1 -1 (Z ... + Z 0 0 ) Z - 2Z ml3 s22 sl2 ml2 Jml2 Js22 Z s l 2 2 Zml2' Zmi2 Z B 2 2 ( Zml3 + " Z s22 ) (3.48) -1 -1 Zml3 + Zs22 ml2 2Z nil2 J s l 2 Js22 Jsl2 -Z sl2 Js22 J s l 2 2Z m!2 Zml2 + Zs22 Jml2 Zml2 Zs22 ( Zml3 + Z s 2 2 ) (3.49) The r a t i o of currents i n sheath to core of phase B i s calculated and plotted as a function of frequency i n Figure3.4. The r e s u l t shows that at frequencies above 1 kHz as i n l i g h t n i n g surges, a l l current through the core w i l l return through i t s own sheath. Thus, one can conclude that the earth return current component i s not important and therefore the mutual 73 c o u p l i n g between cables can be i g n o r e d . c. Sheath current return c h a r a c t e r i s t i c s f o r 3 cable system with current i n a l l 3 cores For a case when currents flox^ i n a l l the three cores of the three SFg cables, the return current through sheath w i l l change accordingly. This suggests that such s i t u a t i o n s must also be investigated to deduce the mutual coupling e f f e c t among cables. Using the same equations as derived i n Equations (3.40) to (3.42), one can now put in currents i n the 3 cores by assuming and i c 2 - i . o IR i c 3 = 1.0 /120° i = 1.0 /-120' c l ' (3.50) (3.51) • (3.52) Rewriting equations (3.40) to (3.42) as Z S 2 2 \l + Zml2 ^ 2 + Zml3 ^ 3 = ' Z s l 2 \l " Zml2 ^ 2 " Zml3 ^ 3 = h <3-53) Zml2 \l + Zs22 S 2 + V ? ^ 3 " " Z,12 S i " Z s l 2 ±c2 " Zml2 S 3 = A2 ( 3 " 5 4 ) Zml3 S i + Zml2 S 2 + Zs22 S 3 = " Zml3 S i " Zml2 S 2 ~ Z s l 2 S 3 = A3 ( 3 ' 5 5 ) Defining the determinant T as Zs22 Zml2 Zml3 Zml2 Zs22 Zml2 Zml3 Zml2 Zs22 Zs232 + 2 Zml2 ' Zml3 " Zs22 ( Zml3 + 2 Z m 1 2 > (3.56) 74 We can then obtain the current through the three i n d i v i d u a l sheaths as' 2 2 2 i s l = A l Z s 2 2 t A3 Zml2 + A 2 Z m 1 2 Z m l 3 " A3 Zs22 Zml3 ~ A2 Zml2 Z s22 " A l Z m l 2 T (3.57) 2 2 i 2 = A2 Zs22 + A l Z m l 2 Zml3 + A3 Zml2 Zml3 ~ A2 Zml3 ~ A3 Zml2 Zs22 " A l Z m l 2 Z s 2 2 T -(3.58) 2 2 2 1S3 " A3 Zs22 + A2 Zrql2 Zral3 + A l Z r a l 2 " A l Z m l 3 Zs22 " A2 Zml2 Zs22 " A3 Zml2 T (3.59) After s u b s t i t u t i n g the conditions, for the 3 phase currents from equations (3.50) to (3.52) into equations (3.57) to (3.59), one can obtain the return currents through a l l i n d i v i d u a l sheaths. The magnitudes of the sheath currents are also shown i n Figure 3.4.it i s again confirmed here that at frequency above 60 Hz, a l l current flowing from core w i l l return through i t s own sheath. Each core i s completely shielded from the adjacent cores. Thus, the three SF^ buses are completely decoupled from one another and should therefore be represented by si n g l e phases as i n the case of l i g h t n i n g overvoltage propagation. 6. Sheath current return c h a r a c t e r i s t i c s f or substation earth with grounding g r i d network In r e a l i t y i n the substation, the cable sheaths are grounded inside the substation with a grounding network g r i d c o n s i s t i n g of copper bars which are connected across the whole substation. These grounding copper bars serve to reduce s i g n i f i c a n t l y the i n s i d e earth r e s i s t i v i t y of the substation. This suggests that the sheath current return c h a r a c t e r i s t i c s of t h i s reduced earth r e s i s t i v i t y should also be investigated as a reduction 75 i n earth r e s i s t i v i t y w i l l favor more current returning through the earth. The r e s u l t f or the sheath return current as a function of earth r e s i s t i v i t y i s shown i n Figure 3.5. This f i g u r e shows that the sheath return current increases as the earth r e s i s t i v i t y increases. This agrees with the manufacturers and u t i l i t y companies of SFg substations who claim that current returning through sheath inside the substation i s l e s s than 75%. Based upon t h i s c r i t e r i a , a nominal earth r e s i s t i v i t y of 0.3 x 10 ^ftm i s chosen. A f t e r choosing a nominal value for earth r e s i s t i v i t y , the sheath current return c h a r a c t e r i s t i c i s then evaluated as a function of frequencies and depth, as shown in Figure-3.6. Fluctuations in o v e r a l l sheath current r e s u l t s are shown i n Figure 3.6. At about 1 K Hz, the sheath current i s even found to be larger than the core current. This can be explained by the phasor diagram as shown i n Figure 3.7. In Figure 3.7,only m u l t i -cable systems with current i n centre core are shown, but mutli-cable system with currents i n a l l 3 cores would Kralso give i d e n t i c a l r e s u l t s . The present study again confirmed that a l l cores are decoupled from one another above 2 k Hz even for the adverse case of s i g n i f i c a n t l y reduced earth r e s i s t i v i t y inside the substation. It should be noted that the mutual impedance between cores, between sheaths and between corresponding cores and sheaths are a l l assumed to be equal by Wedepohl and Ametani. The s h i e l d i n g e f f e c t of the sheath i s neglected. The v a l i d i t y of t h i s assumption i n cable parameter computations could be the topi c of further research. I t i s of l i t t l e concern for the purpose of t h i s t h e s i s . At the high frequencies encountered i n l i g h t n i n g surge studies, core current always return completely through the sheath. In that case, the magnetic f i e l d becomes zero outside the sheath anyhow. log earth resistivity.*; 6 7 8 9 10 Figure 3.5: Ratio of core,current return through sheath at 60 Hz for d i f f e r e n t earth r e s i s t i v i t i e s . 20 30 40 (10"6ft-m) 77 sheath 120 100 80 60 40 z o sheath-mutual sheath earth core single-cable system three-cable system i „/i s2 c2 (§) : depth or height = .254 m (E) : depth or height = 1 m © : depth or height = 6 m p = 3uft-m 20 log frequency i i 1 1 • 10 100 l k 10k 100k (Hz) Figure 3.6: Ratio of core current return through own sheath for s i n g l e - and three- cable system at reduced earth r e s i s t i v i t y of 3 yft-m i n s i d e substation. 78 .1 Hz ->-i core """so i i more core current return through s o i l than sheath 60 Hz i core :". sheath more core current return through sheath than s o i l l k Hz 10k Hz -> ->-i core i core , - — — — ' 1 s h e a t h -y •+ I i J> i 1 sheath 1 c o r e 1 """sheath " c c o r e •"•core """sheath + """soil Figure 3.7: Phasor diagram of current return through sheath and earth. 7 • Formation of shunt admittance matrix for SFfi cables 79 For a usual 3-phase sing l e core underground cable system, one can b u i l d a 6 x 6 shunt admittance matrix [Y ] to describe the cable system as i . e . Ldx J [Y] [V] rdid~ y l - y l 0 0 0 0 dx s i " y l y l + y 2 0 0 0 0 • dx d i c 2 0 0 y l 0 0 dx I d l s 2 0 0 - y l y l + y 2 0 0 dx d i _ c3 0 0 0 0 y l - y l dx d i . s3 0 0 0 0 - y l y l + y 2 dx c l s i c2 V s2 c3 s3 (3.60) Notice that the off-diagonal submatrix of [Y] are a l l zero due to the fact that the grounded sheaths in between acts as e l e c t r o s t a t i c s h i e l d between cables. For the SF^ cable system as shown i n Figure 3.1, the diagonal submatrix elements are i y.. = iwc, = io) 2T T £ '1 J 1 J o Jin r3 y2 = ^ w c2 = 2 7 r e 0 * Jin where y^ i s admittance due to i n t e r n a l SF^ gas i n s u l a t i o n , and i s admittance due to external sheath i n s u l a t i o n . (N.B. ',Th e external i n s u l a t i o n i s non-existent for the SFg cable.) 80 As has been confirmed by the author i n previous f i n d i n g f or sheath current return c h a r a c t e r i s t i c s , the core should be represented as si n g l e phase. Then, the admittance equation shown i n Equation (3.60) should be reduced to d i C y V = J'l C dx In — r2 y s e l f * V c (3.62) 8. Confirmation of numerical accuracy for cable parameter c a l c u l a t i o n and current return r a t i o s The numerical accuracy of the computation was confirmed when the cable parameters obtained by the developed cable constants program, 32 and by the BPA cable constant program agree co n s i s t e n t l y to more than three s i g n i f i c a n t f i g u r e s . 38 Then, a 500 kV submarine cable was chosen as another t e s t example. In t h i s case, the cable parameters for the submarine cable was f i r s t c a l -culated. The amount of core current returning through the sheath, armour and the sea was obtained by taking into account of zero p o t e n t i a l s on the grounded sheath and the grounded armour. The r a t i o s of magnitudes of core current returning through the sheath ,the armour, and the sea at 60 Hz were obtained as 14%, 87.8% and 5.6% r e s p e c t i v e l y . These agreed to more 48 than two figures to the r e s u l t s of other fi n d i n g s . 9. Single phase representation parameters for multi-phase SF6 cables Since a l l phases of the SF^ cables are decoupled from one another, single phase c a b l e representation for studying l i g h t n i n g overvoltage wave 81 propagation i n SF, cable i s recommended. The s e l f admittance element (y ) " s e l f can be calculated from the simple formula as shown i n equations (3.61) and (3.62), whereas the s e l f impedance matrix element can be calculated from equation (3.1) as d V c l Z i + Z i + Z 1 0 ( i „ + i „) + Z . , ( i . + i J (3.63) — == s l l c l sl2 s i ml2 c2 s2 ml3 c3 s3 dx and 0 = Z s 2 1 i c l + Z s 2 2 i s l + Z m l 2 ( i c 2 + i ^ ) + Z m l 3 ( i c 3 + { y ^ Subtracting Equation (3.64) from (3.63), we get ( Z s l l " Zs21> S i + ( Z s l 2 * Z s 2 2 ) S i dx = ( z s i i " W 1 - TT> " W r r " S i > < 3 - 6 5 > c l c l = s e l f c l where the sheath to core current r a t i o can be obtained from equation (3.49) or equations (3.57) to (3.59). Because a l l core current returns through the sheath at high frequencies, one has i n such condition S i = -1, and Z = Z i , ' sl2 s22 . . c l Substituting t h i s into equation (3.65) or (3.37), one obtains dV dx ( Z s l l " Zs22> S ( 3 ' 6 6 ) After having obtained the s e l f s e r i e s impedance for s i n g l e phase cable as shown i n Figure 3...1, one can then represent the cable by the surge surge impedance Z , and the wave propagation v e l o c i t y v, as given by 82 zsurge / ! s e l f = ^ fl ( 3 > 6 ? ) / y s e l f v = oi / - — = 300 m/ys (3.68) / s e l f • y s e l f One should r e a l i z e that the m e t a l l i c sheath of the SF^ cable always form a very good earth return path to the cable core. The seri e s r e s i s t a n c e i s n e g l i g i b l e compared tp the reactance (See Fig.3,8),Thus, the SF^ cable can be taken to be l o s s l e s s . I t should also be noted that f or such a simple go-return c i r c u i t f o r a co a x i a l cable, the inductance can be given by the simple f o r m u l a ^ as y r L = •—- in — (3.69) - • 2TT r 2 = 0.205 VH/m Consistent r e s u l t s f o r the inductance are obtained by equation (3.65) and (3.69) for frequencies above 10 Hz. Thus, the simple formula i s i n equation (3.69) i s recommended for inductance c a l c u l a t i o n of SF^ cable i n the study of surge propagation c h a r a c t e r i s t i c s . The surge impedance and 7 the wave propagation v e l o c i t y can be then obtained as surge = /1 = A ° .. £ n I 3 . _1_ . £ n I 3 C y - 2TT r 2 2 T T E o r 2 ° 1 „ 3 e , 2 r„ o 4TT 2 r3 = 60 £n — (3.70) r2 where r^ and r 2 are radius of sheath and core respectively. 10 Figure 3.8: Self series inductance, resistance and resistance to reactance r a t i o for SF, cable. 84 and v = /f^ = / — J i n — (3.70) •' T r / y r„ 2T T E r„ r2 y £ o o = 300 m / y s v e l o c i t y of l i g h t i n vacuum 10. Wave propagation i n SF6 cables Wave propagation c h a r a c t e r i s t i c s i n sing l e core SF^ cable can now be modelled by the surge impedance of 61.4 ft,(typically about 60 to 75 ft), and wave propagation v e l o c i t y , ( t y p i c a l l y 300 m / y s ) . A numerical simula-ti o n of overvoltage wave-shape .in the receiving end of a SF^ cable j o i n i n g to a overhead transmission l i n e i s simulated. The r e s u l t i n g voltage i n the open-circuited SF^ cable receiving end r i s e s i n a s t a i r c a s e fashion of diminishing amplitude, to a value of 2 p.u. (See Figure 3.9). This can be explained by using the r e f l e c t i o n (C.) and re f r a c t i o n c o e f f i c i e n t (C ) of the system at the li n e - c a b l e junction and is. 39 the open-circuited cable end re s p e c t i v e l y . For the l i n e - c a b l e junction at A, one has Z 2 = 312, Z1 = 60ft Z - Z (Wave incident from cable) c ^ = z + 7^ = 312 '+" 60 = ( 3 - 7 ± ) (Wave incident from l i n e ) Z 2 = 60, Z = 312 ft 85 Figure'3.9: Overvoltage waveshapes at both ends of SF^ cable j o i n i n g from overhead transmission l i n e . 86 For the open end of the cable, we have 2 1 °° - 60 °° + 60 = 1 and C ,o R 2 x °° + 60 2 (Wave incident from cable, Z = 60, Z = ») Thus, the discr e t e r i s e i n voltage wave shape can be expressed as v = 2 x .32 (1 + C.+ C. 2 + . . .) where each step a d d i t i o n accurs at di s c r e t e time i n t e r v a l s of 2 t r a v e l times. On the other hand, t h i s o v e r a l l r i s e i n overvoltages wave shape also agrees with the general exponential r i s e wave shape i n charging of a capacitor. This i s due to the inherent large s e l f capacitance of cables. The o v e r a l l r i s e i n wave shape can : be sketched by modelling the SFg cable as a lumped capacitor equivalent to the t o t a l capacitance for the length of cable, and ignoring the surge impedance of the cable (See Figure 3.9). 87 CHAPTER 4: CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES. 1. Introduction As the l i g h t n i n g voltage wave t r a v e l s down the overhead transmission l i n e , a high e l e c t r i c f i e l d i s produced on the l i n e conductor surface. When kV the e l e c t r i c f i e l d i n t e n s i t y exceeds the breakdown strength of a i r (^30 /em), i o n i z a t i o n of surrounding a i r molecules takes place. This phenomenon w i l l d i s s i p a t e the unwanted surge energy away from the system and thus reduces the magnitude and i n i t i a l rate of r i s e o? t h e l i g h t n i n g overvoltage. In transient l i g h t n i n g overvoltage studies, several numerical methods 44 have been employed to account for corona e f f e c t s . Brown applied the concept of corona radius to account for the corona envelope produced on the conductor surface. The coronated l i n e capacitances at higher voltages are also obtained '• o 12 by extrapolation. Darveniza also used lower wave propagation v e l o c i t i e s higher voltages and d i f f e r e n t corona correction factor for d i f f e r e n t conduc-tor configuration. However, both methods are not straightforward and are 43 not t o t a l l y successful i n d u p l i c a t i n g f i e l d rest r e s u l t s . Umoto and Hara also transformed the transmission l i n e equation for coronated l i n e s into difference algebraic equations. However, t h i s numerical approach i s not e f f i c i e n t enough. Thus, an e f f i c i e n t and accurate numerical model for corona must be developed to predict the corona attenuation and d i s t o r t i o n character-i s t i c s on l i g h t n i n g overvoltage propagations i n overhead l i n e s . 2. Physical properties of corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s The p h y s i c a l aspects and laws governing the behaviour of corona d i s -charge have been investigated since the beginning of t h i s century. However, most of the investigations and a p p l i c a t i o n s have been l i m i t e d to power frequency steady state or at most to switching transient conditions. From 88 the published f i e l d measurements f o r l i g h t n i n g surges, i t can be observed that the attenuation r e s u l t i n g from corona e f f e c t s i s much larger than that r e s u l t i n g from transmission l i n e s e r i e s resistance losses. The non-linear c h a r a c t e r i s t i c s of the corona discharge can be considered as (see Figure 4.1): a) Corona attenuation loss - From the quadratic law of corona loss pro-41 posed by Peek , the loss (v i^) per un i t length i s proportional to the square of the voltage above the c r i t i c a l corona voltage v i . e . where k = a • /-^r x 10 /m r , h = radius and height of conductor r e s p e c t i v e l y a = Corona loss constant determined experimentally This corona attenuation l o s s can be modelled with a r e s i s t i v e current l o s s i ^ through the corona r e s i s t i v e branch to ground as b) Increase i n shunt capacitance - the retardation of the wave front by 42 corona can be explained by an increase i n shunt capacitance. S k i l l i n g 43 and Umoto suggested that the increase i n shunt capacitance i s proportional to the voltage above the c r i t i c a l voltage V c q , i . e . V C = 2k (1 - (4.3) corona c V where k = a x 10 h c c v 2h a = corona los s constant^determined experimentally c 89 V - V R - v c transmission l i n e Corona shunt capacitance V / C = 2k (1 „ corona c V. 1 + K c R G =kn--~f corona R V rrfn rmr Figure 4.1: Nonlinear corona losses model. 90 This increase i n capacitance can be modelled by : a capacitance branch to ground with the capactive current los s i„ Corona discharge only occur i f the voltage i s greater or equal to the c r i t i c a l corona voltage, and i f the voltage increase with time, i . e . j 3v v £ v , and — > o. co 3t This i s due to the fact that, when the voltage begins to decrease, the space charge c o n s i s t i n g of heavy ions i n the i o n i z a t i o n region remains p r a c t i c a l l y constant i n magnitude and p o s i t i o n during a short period of time. This slow d i f f u s i o n of ions r e s u l t s i n l i t t l e energy loss i n the case of decreasing voltage conditions even when v > V C Q . 3. Transmission l i n e equations f o r coronated l i n e s . The corona phenomena can now be described by the modified l i n e equations. With the introduction of d i g i t a l computers, these phenomena can be studied accurately by solving the equations describing the electromagnetic wave propagations taking corona into account as follows: = £ + ( i - ^ S + v1-'^2-' <4-6> extra shunt Extra shunt capacitance conductance due to due to corona corona 43 45 Umoto and Inoue solved the above equations by the d i f f e r e n c e method. The l i n e equations (4.5) and (4.6) are transformed into algebraic equations of small increments of distance, Ax, and time, At. However, t h i s method i s not e f f i c i e n t to implement into the d i g i t a l computer as the method requires Ax to be as small as 7 m r when using At = .01 ys. 4. Solution of l i n e equation by compensation method with trapezoidal rules The l i n e equations (4.5) and (4.6) with corona losses can be solved by the compensation method. In t h i s method, the l i n e equations are f i r s t solved without the extra corona terms. The Bergeron's method using t r a v e l l i n g wave technique together with modal ahalysis(See Chapter 2) i s a p p l i e d . Then, the corona losses can be treated as non-linear shunt branches connected to ground,. .< The trapezoidal r u l e can then applied to obtain the t o t a l current los s of the corona phenomena. By applying the trapezoidal rule of l i n e a r i n t e r p o l a t i o n to the corona r e s i s t i v e branch to ground, we have j as shown i n Figure 4.2, ± - e - v t + A t = i W + ( v t " i V where d = + — (4.7) v — v t + At t as d = slope of graph at time t . Also, d can be obtained by considering the equation (4.2) 92 Current i ( v ) ( Voltage v Figure 4.2: Linear i n t e r p o l a t i o n for resistance corona branch. 8v/9t=v voltage v or current i Figure 4.3: Linear i n t e r p o l a t i o n f or capacitive corona branch. 93 ( v - v )' t CO t R v. or co k^ v t + k„ — - 2k„ v R v ~"R co d i d = dv v 2 K. kR v 2 (4.8) Thus, we eventually have 1 . , ,v - 1 i v V t . + At " d \ + At + ( fc d ° (4.9) \ ' i t + At + V o (4.10) where v = v - v i , ( known from past h i s t o r y at time t) o t d t and ^R = a" (known from past h i s t o r y at time t) S i m i l a r l y , since the corona capacitive branch current loss i s given by 2 k a c , , 9v x = (v - v ) — v co 3t or 9v v i 3t 2k (v - v ) c CO f ( v , i ) Applying l i n e a r i n t e r p o l a t i o n of the 2 variables ( i . e . from f i r s t term of Taylors' s e r i e s ) , we have as shown i n Figure 4.3 f ( v , i ) = f ( v , i ) t+At + 3v (v t+At \ , 3f ' Vt> + 31 ^t+At " ± t ) (4.11) 94 3f 3v i (v v ) - v 2k t c C O - X , V C O 2 k c <vt - v c o ) 2 (4.12) and 3f 3i 2kj;v - v ) t * co 2H\ - \o> (4.13) Thus, we obtain f ( v , i ) t+At 3v 3t t+At v — v t+At t " A t v. 2k(v - v ) v t+At / t ^ t co / Re-arranging equation (4.14) w i l l give the l i n e a r i z e d equation as Vt+At = R c S+At + V l (4.15) where R = 1 + v i * . . co t At 2k (v - v )2 C t C O * v At t (known from past 2k (v - v ) , . „ . c t co hxstory) and 1 + v K' A«-co t At 2k (v - v )2 c t co . . V X V At co t t ( V t + 2k (v - v )2> from c t co past hxstory) Combining equations (4.10) and (4.15) for the voltages and currents in both corona r e s i s t i v e and ca p a c i t i v e branches by taking into account v = v_. = v c R and i = i + i„ c R we can obtain ( ^ + ^ ) v - ( i c + i R ) + ( ^ + ^ ) of or v = R ' i + k' (4.17) R c *R where R' and k* = R c + R k *R V l + R c Vo R Having the corona loss branches represented by a l i n e a r model as described in equation (4.17), the compensation method can then be applied to solve the transmission l i n e equations including corona losses. In the compensation method, the transmission l i n e i s f i r s t reduced to a Thevinin equivalent (See Figure 4.4) and i s described by vv= V Q + A2±, (4.18) where i s a negative number. Then, t h i s equation i s solved simultaneously with the l i n e a r i z e d equation for corona l o s s , as in equation (4.17). Thus, the r e s u l t i n g corona voltage and discharge current can be obtained as A m (m i s ground) A : Thevinin equivalent network for transmission l i n e without corona losses B : Nonlinear corona losses model v = R' i + k' Voltage v. v corona «• v =v +A i • » v k m o 2 Current i km i corona Figure 4.4: Compensation method f o r non-linear corona model 1corona - R' - A. (4.19) R'v - A-k* A o 2 and v corona • R1 - A. (4.20) 5. Influence on corona by adjacent sub-conductors i n the same bundle Extra-high voltage phase conductors are designed to consist of several sub-conductors bundled together in order to reduce corona losses. The e l e c t r i c f i e l d on a sub-conductor surface i s affected appreciably by the adjacent sub-conductors i n the same bundles. The corona phenomenon i s consequently influenced. The e l e c t r i c f i e l d on the sub-conductor surface due to the sub-46 conductor i t s e l f i s given by max 2 IT e r o , Qr-= charge/length cv 2 IT e r o (4.21) where c = e f f e c t i v e capacitance/length v = voltage of conductor r = radius of sub-conductor However, for a bundled conductor with 4 i n d i v i d u a l sub-conductors, the maximum e l e c t r i c f i e l d i s given by^(See Figure 4.5) max Q1 ,- 1 2TT e + s72 + 2 -• s i n 45 u) -21 •- -2-TT e r o (4.22) 98 \ a x - -2^ tT ( 1 + vfs } C V „, e f f e c t i v e where Q = ^ Figure 4.5: C r i t i c a l voltage c a l c u l a t i o n by evaluation of maximum e l e c t r i c f i e l d on a 4-conductor bundle. A f t e r the maximum e l e c t r i c f i e l d on the conductor surface i s obtained, the c r i t i c a l voltage for corona discharge can be computed by 6 equating the maximum e l e c t r i c f i e l d to 30 kV/cm or 3 x 10 v/m, . the e l e c t r i c breakdown strength i n a i r . A t y p i c a l c r i t i c a l voltage f o r a single conductor has been found to be 277 kV, and that for a 4-conductor bundle to be 558 kV. 6. Influence on corona by adjacent phase conductors Since the conductors i n each phase are mutually coupled to one another, voltages are always induced i n the adjacent conductors. Thus the maximum e l e c t r i c f i e l d on the conductor surface i s affected. However, due to the design of transmission l i n e s for extra high voltage l e v e l s , separating between phase conductors are u s u a l l y large compared with radius of i n d i v i d u a l conductors. This e f f e c t u sually change the o v e r a l l c r i t i c a l overvoltages by l e s s than 10%. But t h i s change i n c r i t i c a l voltage produces n e g l i g i b l e e f f e c t s on the o v e r a l l corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s on overvoltage wave (See Figure 4.6). 7. Optimal lumping locations and number of corona branch legs The equations with corona phenomenon i s now solved by the dbmbensation method with the corona l o s s legs lumped at a few places along the transmission•-• l i n e . However, the optimal locations and optimal number of lumped elements has to be determined. At f i r s t , 20 corona loss branches 70 m apart from one another were lumped between f i v e transmission towers. Then, a separation of 350 m between the corona loss branches was used. This increase i n separa-ti o n increased the deviation of the predicted wave shape from f i e l d measure-ments' appreciably (See Figure 4.7), from about 5 to 10%. This suggests that f i e l d measurements v ± t ± c a i = 303-kV (include e f f e c t of. c r i xca adjacent phase conductors)) v . . 277 kV (neglect e f f e c t of c r x t i c a l adjacent phase conductors) Figure 4.6: E f f e c t of adjacent phase conductors on corona losses. Figure 4.7: E f f e c t of lumping distances on corona. 102 the optimal separation should be about 70 m. The f i e l d measurement for a 4-conductor bundled was then simulated. In using the corona loss constants (for case of 1-conductor bundle) a = 30 c a- = 10 x 10 6 s l i g h t l y higher overvoltages were obtained. Then, a new set of corona constants (for case of 4-conductors bundle) a = 30 c a = 20 x 10 6 was used to give r e s u l t s consistent with those from f i e l d measurements (See Figure 4.8). F i n a l l y , the negative impulse overvoltage was also simulated f o r the 4-conductor bundle case. The corona loss i n t h i s case was found to be much less than the p o s i t i v e Impulse case. The corona los s constants were determined to be a = 15 c a = 10 x 10 6 With these sets of corona constants, the f i e l d t e s t measurement was again r e p l i c a t e d c l o s e l y (See Figure 4.9). 8. Overall numerical modelling f o r corona e f f e c t s The f i e l d t e s t r e s u l t s of corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s on a 500 kV te s t l i n e were r e p l i c a t e d by the method Overvoltages(kV) Figure 4.8: P o s i t i v e impulse on 4- conductor bundle." Overvoltages(kV) 2000 FDUR-CONDUCTOR BUNDLE 1 2 3 4 5 6 f i e l d measurements a =15, a =10xl0 6 c G r a =15, a =5x10 c G Figure 4.9: Negative impulse on 4T conductor bundle. 105 developed e a r l i e r . This method examined corona c h a r a c t e r i s t i c s i n both single and bundled conductor l i n e s . From the performed study, one can conclude that the e f f e c t s of bundling of conductors i s e f f i c i e n t i n increas-ing c r i t i c a l corona voltage. Furthermore, influence of adjacent phase conductors i s n e g l i g i b l e on corona e f f e c t s . Thus, i t i s concluded that s i n g l e phase l i n e representation i s s u f f i c i e n t f o r corona studies. F i n a l l y , i t i s determined that separation between the corona los s leg can be lumped at 70 m without s a c r i f i c i n g a loss of accuracy on the predicted coronated waveform. A. reduction i n distance between corona legs w i l l not improve the accuracy of the simulated r e s u l t s . It should be noted that l i g h t n i n g strokes w i l l r a r e l y h i t more than one conductor at one time; thus corona phenomena have only been included for one conductor i n t h i s t h e s i s , rather than for a l l three phases simultaneously. CHAPTER 5: CONCLUSIONS 106 The attentuation and d i s t o r t i o n of l i g h t n i n g overvoltage waves on multi-phase transmission l i n e s and multi-phase single core SF^ cables i n compressed SF^ gas-insulated substations was studied. Corona e f f e c t s of l i g h t n i n g overvoltages on overhead l i n e s were also investigated. Available f i e l d t e s t r e s u l t s for corona e f f e c t s were duplicated to within 5% accuracy. Results obtained with the techniques developed by the author are 21 useful f or l i g h t n i n g i n s u l a t i o n co-ordination studies and other re l a t e d ,. 13,14,40 ^ , i t J studies . xhe l i g h t n i n g surge wave front can be calculated at any l o c a t i o n inside the substation, eg., inside the SF^ bus or at the trans-o former terminal. Based on the studies described i n the thesis the following recommendations are made for future i n s u l a t i o n co-ordination design studies: 1. Multi-phase untransposed l i n e s can be represented by single-phase l i n e models using s e l f parameters calculated at a high frequency of approximately 1 M Hz (See Table 2.4). Series resistance should be ignored. Frequency dependent e f f e c t s are not important f o r propa-gation over distances l e s s than 2 km. 2. Corona e f f e c t s are important i n reducing the magnitude and rate of r i s e of the incoming l i g h t n i n g overvoltage surge. E f f i c i e n t s o l u t i o n techniques using compensation methods are developed to solve the non-l i n e a r corona attenuation and d i s t o r t i o n phenomenon. 3. Multi-phase SF^ si n g l e core cables can be represented by s i n g l e phase cable models. Series resistance can be ignored. Cable parameter can be obtained with the simple formula for a go-return c i r c u i t f o r a coaxial cable with s u f f i c i e n t accuracy. 107 APPENDIX A: SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY. This section shows that the core current return c h a r a c t e r i s t i c s through the sheath for the SF^ cable could be obtained by a d i f f e r e n t approach. From the Maxwell's equations i n a conducting medium, we h a v e ^ V x E = -jwu H (A.l) V x H = jtoeE + aE = aE, for good conductors (A.2) where a i s conductivity of medium. From equations (A.l) and (A.2), we can get 2 2 V x V x E = V ( V « E ) - V E = - V E (for homogeneous medium) = -joiyV x H = -jwyaE = -m2E where m = /jtoya = ^ ^ • /u>ya This equation i s i d e n t i c a l to the d i f f u s i o n equation with solutions — = E = E e - m Z , (A.3) a x o = E e-V"2~ Z • e - V ~ T Z (A.4) o = E e - j Z / 6 • e " j Z / 6 (A.5) o 108 where 6 = ~-. = - 1 7= skin depth Thus, the tangential e l e c t r i c f i e l d E or the tangential current density J x w i l l be attenuated by ^ = -368 when the depth of penetration Z equals to the skin depth. For aluminum, we have the skin depth 6 as <5 = 7 = = (A.6) /irf (4ITX10-7) (3-8x10/) 8-1 ,— cm Therefore, for frequency above 1 kHz, the e l e c t r i c f i e l d i s e s s e n t i a l l y attenuated and n e g l i g i b l e f l u x outside the sheath. Thus, since character-i s t i c frequencies of l i g h t n i n g strokes exceeds 1 kHz, the above r e s u l t s i n d i c a t e that each phase of the cable i s decoupled from other phases as was shown previously i n Chapter 3. A f t e r the tangential current density f o r one medium i s obtained by equations (A.3) to (A.5), the tangential current density for another medium on the boundary to the f i r s t medium can be obtained by E l t = E 2 t °1 J l t = ~ 2 J 2 t Thusm the t o t a l current flowing i n d i f f e r e n t components of the cable system can be obtained by I = /JdA 109 BIBLIOGRAPHY 1. W. Diesendorf, 'Insulation coordination i n high voltage e l e c t r i c power systems', Butterworth Cp. London 1974. 2. M. 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Inoue, 'High voltage t r a v e l l i n g waves with corona discharge on bundle conductors', presented to the IEEE 1978 Winter Meeting A78 170-3. 46. W. Hayt, 'Engineering Electromagnetics', McGraw , H i l l , N.Y. 1*974. 47. H.W.Dommel, W.S.Meyer, 'Computation of electromagnetic t r a n s i e n t s ' , Proc. IEEE vol.62, pp.983-993, July 1974. 48. H.W.Dommel, priv a t e communication July 1979. 49. G. Strang, 'Linear algebra and I t s applications',Academic Press N.Y. 1976. 112 50. ,'A.T.Adams, ' Electromagnetics for engineers', Ronald Press Company, New York, N.Y. 1971. 51." G.L. Ford, E.P.Dick, ' Study of fault-produced oyervoltages as-rel a t e d to grounding p r a c t i s e f o r SF -insulated substations', Canadian E l e c t r i c a l Association Contract No. 071T101, Sept. 1979. 52. ' Lightning protection design guide book f o r power stations and substations', Central Research I n s t i t u t e of E l e c t r i c Power Industry CRIEPI Report No. 175034, Tokyo, Japan 1976. (in Japanese)
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Lightning surge propagation in overhead lines and gas insulated bus-ducts and cables Lee, Kai-Chung 1980
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Title | Lightning surge propagation in overhead lines and gas insulated bus-ducts and cables |
Creator |
Lee, Kai-Chung |
Publisher | University of British Columbia |
Date Issued | 1980 |
Description | The propagation characteristics of lightning surges in compressed SF₆ gas insulated power substation was studied using an electromagnetic transients program. Numerical models were developed to represent the behaviour of different system components especially under lightning over-voltage conditions. The characteristics of lightning surge propagation in overhead multi-phase untransposed transmission lines was analysed first. Modal analysis, together with special rotation techniques to fit time domain solutions were then used to simulate the wave propagation in multi-phase untransposed line in an electromagnetic transients program. Non-linear voltage-dependent corona attenuation and distortion phenomena were also investigated. Available field test results could be duplicated to within 5%. The characteristics of lightning surge propagation in multi-phase single-core SF₆ cables was studied next. A program was developed to obtain the cable parameters for typical cable configurations. The amount of core current returning through its own sheath and through the earth were computed to illustrate the single phase cable representation for wave propagation in single core SF₆ cables. |
Subject |
Lightning Electric power transmission -- Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065488 |
URI | http://hdl.handle.net/2429/22543 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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